{"id":207,"date":"2019-08-20T17:04:07","date_gmt":"2019-08-20T21:04:07","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/series-and-their-notations\/"},"modified":"2022-06-01T10:39:40","modified_gmt":"2022-06-01T14:39:40","slug":"series-and-their-notations","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/series-and-their-notations\/","title":{"raw":"Series and Their Notations","rendered":"Series and Their Notations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n \t<li>Use summation notation.<\/li>\n \t<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\n \t<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\n \t<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\n \t<li>Solve annuity problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137612041\">A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.<\/p>\n\n<div id=\"fs-id1165137405637\" class=\"bc-section section\">\n<h3>Using Summation Notation<\/h3>\n<p id=\"fs-id1165137788981\">To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.<\/p>\n\n<div id=\"fs-id1165135317537\" class=\"unnumbered aligncenter\">[latex]3+7+11+15+19+...[\/latex]<\/div>\n<p id=\"fs-id1165137639252\">The [latex]n\\text{th }[\/latex]partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation[latex]\\text{ }{S}_{n}\\text{ }[\/latex]represents the partial sum.<\/p>\n\n<div id=\"eip-901\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{1}=3\\\\ {S}_{2}=3+7=10\\\\ {S}_{3}=3+7+11=21\\\\ {S}_{4}=3+7+11+15=36\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137653088\"><strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <span class=\"no-emphasis\">sigma<\/span>, [latex]\\text{\u03a3},[\/latex] to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.<\/p>\n<span id=\"fs-id1165137432096\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154910\/CNX_Precalc_Figure_11_04_001n.jpg\" alt=\"Explanation of summation notion as described in the text.\"><\/span>\n<p id=\"fs-id1165137758894\">If we interpret the given notation, we see that it asks us to find the sum of the terms in the series[latex]\\,{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5.\\,[\/latex] We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.<\/p>\n\n<div id=\"fs-id1165137726334\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{1}=2\\left(1\\right)=2\\end{array}\\hfill \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137678008\">We can find the sum of the series by adding the terms:<\/p>\n\n<div id=\"fs-id1165135481276\" class=\"unnumbered aligncenter\">[latex]\\sum _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/div>\n<div id=\"fs-id1165137726612\" class=\"textbox key-takeaways\">\n<h3>Summation Notation<\/h3>\n<p id=\"fs-id1165137471824\">The sum of the first[latex]n[\/latex]terms of a <strong>series <\/strong>can be expressed in summation notation as follows:<\/p>\n\n<div id=\"fs-id1165137938486\" class=\"unnumbered aligncenter\">[latex]\\sum _{k=1}^{n}{a}_{k}[\/latex]<\/div>\n<p id=\"fs-id1165134109690\">This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from [latex]k=1[\/latex] to [latex]k=n.[\/latex]<\/p>\n<p id=\"eip-419\">[latex]k\\,[\/latex] is called the index of summation, 1 is the lower limit of summation, and [latex]n[\/latex] is the upper limit of summation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137732268\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1165135496558\"><strong>Does the lower limit of summation have to be 1?<\/strong><\/p>\n<p id=\"fs-id1165137727904\"><em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em><\/p>\n\n<\/div>\n<div class=\"precalculus howto\">\n<p id=\"fs-id1165137761194\"><strong>Given summation notation for a series, evaluate the value.<\/strong><\/p>\n\n<ol id=\"fs-id1165137731973\" type=\"1\">\n \t<li>Identify the lower limit of summation.<\/li>\n \t<li>Identify the upper limit of summation.<\/li>\n \t<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\n \t<li>Add to find the sum.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1165135209801\">\n<div id=\"fs-id1165137466005\">\n<h3>Using Summation Notation<\/h3>\n<p id=\"fs-id1165137834859\">Evaluate[latex]\\sum _{k=3}^{7}{k}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div>\n<div id=\"fs-id1165137464424\" class=\"unnumbered aligncenter\">\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137472839\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137472839\"]\n<p id=\"fs-id1165137472839\">According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7.[\/latex] We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6\\text{,}\\,[\/latex] and [latex]7[\/latex] into the function [latex]{k}^{2}.[\/latex] We add the terms to find the sum.<\/p>\n\n<div id=\"fs-id1165137464424\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\sum _{k=3}^{7}{k}^{2}\\hfill &amp; ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill &amp; =9+16+25+36+49\\hfill \\\\ \\hfill &amp; =135\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482825\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137702091\">\n<p id=\"fs-id1165137806341\">Evaluate[latex]\\sum _{k=2}^{5}\\left(3k\u20131\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137658521\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137658521\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137658521\"]\n<p id=\"fs-id1165137589299\">38<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137480338\" class=\"bc-section section\">\n<h3>Using the Formula for Arithmetic Series<\/h3>\n<p id=\"fs-id1165137557625\">Just as we studied special types of sequences, we will look at special types of series. Recall that an <span class=\"no-emphasis\">arithmetic sequence<\/span> is a sequence in which the difference between any two consecutive terms is the <span class=\"no-emphasis\">common difference<\/span>,[latex]d.[\/latex] The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:<\/p>\n\n<div id=\"eip-id1165137653029\" class=\"unnumbered\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}\u2013d\\right)+{a}_{n}.[\/latex]<\/div>\n<p id=\"fs-id1165135176760\">We can also reverse the order of the terms and write the sum as<\/p>\n\n<div id=\"eip-id1165137874798\" class=\"unnumbered\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}\u2013d\\right)+\\left({a}_{n}\u20132d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}.[\/latex]<\/div>\nIf we add these two expressions for the sum of the first [latex]n[\/latex]terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.\n<div id=\"eip-id1165135169564\" class=\"unnumbered\">[latex]\\frac{\\begin{array}{l}\\,\\,\\,\\,\\,\\,{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}\u2013d\\right)+{a}_{n}\\hfill \\\\ +\\,\\,{S}_{n}={a}_{n}+\\left({a}_{n}\u2013d\\right)+\\left({a}_{n}\u20132d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}\\hfill \\end{array}}{2{S}_{n}=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165134148512\">Because there are [latex]n[\/latex] terms in the series, we can simplify this sum to<\/p>\n\n<div id=\"eip-id1165137726382\" class=\"unnumbered\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right).[\/latex]<\/div>\n<p id=\"fs-id1165135505002\">We divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.<\/p>\n\n<div id=\"eip-id1165137447374\" class=\"unnumbered\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/div>\n<div id=\"fs-id1165135506407\" class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of the First <em>n<\/em> Terms of an Arithmetic Series<\/h3>\n<p id=\"fs-id1165134040601\">An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first [latex]n[\/latex] terms of an arithmetic sequence is<\/p>\n\n<div id=\"fs-id1165137741027\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135255874\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137529710\"><strong>Given terms of an arithmetic series, find the sum of the first [latex]n[\/latex] terms.<\/strong><\/p>\n\n<ol id=\"fs-id1165135646144\" type=\"1\">\n \t<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}.[\/latex]<\/li>\n \t<li>Determine [latex]n.[\/latex]<\/li>\n \t<li>Substitute values for [latex]{a}_{1}\\text{, }{a}_{n}\\text{,}\\,[\/latex] and [latex]\\,n\\,[\/latex] into the formula [latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}.[\/latex]<\/li>\n \t<li>Simplify to find [latex]{S}_{n}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135191078\">\n<div id=\"fs-id1165135191081\">\n<h3>Finding the First <em>n<\/em> Terms of an Arithmetic Series<\/h3>\n<p id=\"fs-id1165135159870\">Find the sum of each arithmetic series.<\/p>\n\n<ol id=\"fs-id1165135159873\" type=\"a\">\n \t<li>[latex]\\text{5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32}[\/latex]<\/li>\n \t<li>[latex]\\text{20 + 15 + 10 +\u2026+ \u221250}[\/latex]<\/li>\n \t<li>[latex]\\sum _{k=1}^{12}3k-8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135571638\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135571638\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135571638\"]\n<ol id=\"fs-id1165135571640\" type=\"a\">\n \t<li>\n<p id=\"fs-id1165137665556\">We are given [latex]{a}_{1}=5[\/latex] and [latex]\\,{a}_{n}=32.[\/latex]<\/p>\n<p id=\"fs-id1165135404253\">Count the number of terms in the sequence to find [latex]n=10.[\/latex]<\/p>\n<p id=\"fs-id1165135417695\">Substitute values for [latex]\\,{a}_{1},{a}_{n}\\text{\\hspace{0.17em},}[\/latex] and [latex]n[\/latex] into the formula and simplify.<\/p>\n\n<div id=\"eip-id1165137748560\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\end{array}\\hfill \\\\ {S}_{10}=\\frac{10\\left(5+32\\right)}{2}=185\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>\n<p id=\"fs-id1165137832192\">We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50.[\/latex]<\/p>\n<p id=\"fs-id1165135186923\">Use the formula for the general term of an arithmetic sequence to find [latex]n.[\/latex]<\/p>\n\n<div id=\"eip-id1165137806527\" class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ -50=20+\\left(n-1\\right)\\left(-5\\right)\\hfill \\\\ -70=\\left(n-1\\right)\\left(-5\\right)\\hfill \\\\ \\,\\,\\,\\,14=n-1\\hfill \\\\ \\,\\,\\,\\,15=n\\hfill \\end{array}[\/latex]<\/div>\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}\\,n[\/latex] into the formula and simplify.\n<div><\/div>\n<div id=\"eip-389\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\end{array}\\hfill \\\\ {S}_{15}=\\frac{15\\left(20-50\\right)}{2}=-225\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>\n<p id=\"fs-id1165135175094\">To find [latex]{a}_{1},\\,[\/latex]substitute [latex]k=1[\/latex] into the given explicit formula.<\/p>\n\n<div id=\"eip-id1165135528406\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{k}=3k-8\\hfill \\\\ \\text{ }{a}_{1}=3\\left(1\\right)-8=-5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137627558\">We are given that [latex]n=12.[\/latex] To find [latex]{a}_{12},\\,[\/latex]substitute [latex]k=12[\/latex] into the given explicit formula.<\/p>\n\n<div id=\"eip-532\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }{a}_{k}=3k-8\\hfill \\\\ {a}_{12}=3\\left(12\\right)-8=28\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137641682\">Substitute values for [latex]{a}_{1},{a}_{n},[\/latex] and [latex]n[\/latex] into the formula and simplify.<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ {S}_{12}=\\frac{12\\left(-5+28\\right)}{2}=138\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-32\">Use the formula to find the sum of each arithmetic series.<\/p>\n\n<div id=\"fs-id1165134032286\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_02\">\n<div id=\"fs-id1165134092405\">\n<p id=\"eip-id1926809\">[latex]\\text{1}\\text{.4 + 1}\\text{.6 + 1}\\text{.8 + 2}\\text{.0 + 2}\\text{.2 + 2}\\text{.4 + 2}\\text{.6 + 2}\\text{.8 + 3}\\text{.0 + 3}\\text{.2 + 3}\\text{.4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134226148\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134226148\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134226148\"]\n<p id=\"fs-id1165134226150\">[latex]\\text{26}\\text{.4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_03\">\n<div id=\"fs-id1165135198570\">\n<p id=\"eip-id1862925\">[latex]\\text{13 + 21 + 29 + }\\dots \\text{+ 69}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135198577\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135198577\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135198577\"]\n<p id=\"fs-id1165135198579\">[latex]\\text{328}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_04\">\n<div id=\"fs-id1165137651940\">\n<p id=\"eip-id2303118\">[latex]\\sum _{k=1}^{10}5-6k[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137874274\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137874274\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137874274\"]\n<p id=\"fs-id1165137874276\">[latex]\\text{\u2212280}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135208460\">\n<div id=\"fs-id1165135208462\">\n<h3>Solving Application Problems with Arithmetic Series<\/h3>\n<p id=\"fs-id1165134382167\">On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134382173\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134382173\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134382173\"]\n<p id=\"fs-id1165134382175\">This problem can be modeled by an arithmetic series with[latex]\\,{a}_{1}=\\frac{1}{2}\\,[\/latex]and[latex]\\,d=\\frac{1}{4}.\\,[\/latex]We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8\\text{,}[\/latex] and we are looking for[latex]\\,{S}_{8}.\\,[\/latex]To find [latex]{a}_{8},[\/latex] we can use the explicit formula for an arithmetic sequence.<\/p>\n\n<div id=\"eip-id1165135512530\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{n}={a}_{1}+d\\left(n-1\\right)\\end{array}\\hfill \\\\ {a}_{8}=\\frac{1}{2}+\\frac{1}{4}\\left(8-1\\right)=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137771149\">We can now use the formula for arithmetic series.<\/p>\n\n<div id=\"eip-id1165137653366\" class=\"unnumbered\">[latex]\\begin{array}{l} {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ \\text{ }{S}_{8}=\\frac{8\\left(\\frac{1}{2}+\\frac{9}{4}\\right)}{2}=11\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137673659\">She will have walked a total of 11 miles.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673666\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_05\">\n<div id=\"fs-id1165137734534\">\n<p id=\"fs-id1165137734536\">A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137660095\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137660095\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137660095\"]\n<p id=\"fs-id1165137660097\">$2,025<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660105\" class=\"bc-section section\">\n<h3>Using the Formula for Geometric Series<\/h3>\n<p id=\"fs-id1165135188612\">Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a <strong>geometric series<\/strong>. Recall that a <span class=\"no-emphasis\">geometric sequence<\/span> is a sequence in which the ratio of any two consecutive terms is the <span class=\"no-emphasis\">common ratio<\/span>, [latex]\\,r.\\,[\/latex]We can write the sum of the first [latex]n[\/latex] terms of a geometric series as<\/p>\n\n<div class=\"unnumbered\">[latex]{S}_{n}={a}_{1}+r{a}_{1}+{r}^{2}{a}_{1}+...+{r}^{n\u20131}{a}_{1}.[\/latex]<\/div>\n<p id=\"fs-id1165135504988\">Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first[latex]\\,n\\,[\/latex]terms of a geometric series. We will begin by multiplying both sides of the equation by[latex]\\,r.\\,[\/latex]<\/p>\n\n<div id=\"eip-41\" class=\"unnumbered aligncenter\">[latex]r{S}_{n}=r{a}_{1}+{r}^{2}{a}_{1}+{r}^{3}{a}_{1}+...+{r}^{n}{a}_{1}[\/latex]<\/div>\n<p id=\"fs-id1165134172597\">Next, we subtract this equation from the original equation.<\/p>\n\n<div id=\"fs-id1165135704813\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\frac{\\begin{array}{l}\\text{ }{S}_{n}={a}_{1}+r{a}_{1}+{r}^{2}{a}_{1}+...+{r}^{n\u20131}{a}_{1}\\hfill \\\\ -r{S}_{n}=-\\left(r{a}_{1}+{r}^{2}{a}_{1}+{r}^{3}{a}_{1}+...+{r}^{n}{a}_{1}\\right)\\hfill \\end{array}}{\\left(1-r\\right){S}_{n}={a}_{1}-{r}^{n}{a}_{1}}\\end{array}[\/latex]<\/div>\nNotice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n},[\/latex] divide both sides by [latex]\\left(1-r\\right).[\/latex]\n<div id=\"fs-id1165137871953\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/div>\n<div id=\"fs-id1165137533230\" class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of the First <em>n<\/em> Terms of a Geometric Series<\/h3>\n<p id=\"fs-id1165137863800\">A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first[latex]\\,n\\,[\/latex]terms of a geometric sequence is represented as<\/p>\n\n<div>[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137602804\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<strong>Given a geometric series, find the sum of the first <em>n<\/em> terms.<\/strong>\n<ol id=\"fs-id1165137698300\" type=\"1\">\n \t<li>Identify[latex]\\,{a}_{1},\\,r,\\,\\text{and}\\,n.[\/latex]<\/li>\n \t<li>Substitute values for[latex]\\,{a}_{1},\\,r,[\/latex] and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\frac{{a}_{1}\\left(1\u2013{r}^{n}\\right)}{1\u2013r}.[\/latex]<\/li>\n \t<li>Simplify to find [latex]{S}_{n}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135196954\">\n<div id=\"fs-id1165135196957\">\n<h3>Finding the First <em>n<\/em> Terms of a Geometric Series<\/h3>\nUse the formula to find the indicated partial sum of each geometric series.\n<ol id=\"fs-id1165137736582\" type=\"a\">\n \t<li>[latex]{S}_{11}[\/latex]for the series[latex]\\text{ 8 + -4 + 2 + }\\dots [\/latex]<\/li>\n \t<li>[latex]\\underset{k=1}{\\overset{6}{{\\sum }^{\\text{\u200b}}}}3\\cdot {2}^{k}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135264856\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135264856\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135264856\"]\n<ol id=\"fs-id1165135264859\" type=\"a\">\n \t<li>\n<p id=\"fs-id1165137732247\">[latex]{a}_{1}=8,[\/latex] and we are given that [latex]n=11.[\/latex]<\/p>\n<p id=\"fs-id1165135255878\">We can find [latex]r[\/latex] by dividing the second term of the series by the first.<\/p>\n\n<div id=\"fs-id1165135564192\" class=\"unnumbered aligncenter\">[latex]r=\\frac{-4}{8}=-\\frac{1}{2}[\/latex]<\/div>\n<p id=\"fs-id1165137755996\">Substitute values for [latex]{a}_{1}, r, \\text{and} n[\/latex] into the formula and simplify.<\/p>\n\n<div id=\"fs-id1165133347581\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{11}=\\frac{8\\left(1-{\\left(-\\frac{1}{2}\\right)}^{11}\\right)}{1-\\left(-\\frac{1}{2}\\right)}\\approx 5.336\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>\n<p id=\"fs-id1165137673547\">Find [latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.<\/p>\n\n<div id=\"fs-id1165137932596\" class=\"unnumbered aligncenter\">[latex]{a}_{1}=3\\cdot {2}^{1}=6[\/latex]<\/div>\n<p id=\"fs-id1165137656631\">We can see from the given explicit formula that [latex]r=2.[\/latex] The upper limit of summation is 6, so [latex]n=6.[\/latex]<\/p>\n<p id=\"fs-id1165135149030\">Substitute values for [latex]{a}_{1},\\,r,[\/latex] and [latex]n[\/latex] into the formula, and simplify.<\/p>\n\n<div id=\"fs-id1165137400638\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{6}=\\frac{6\\left(1-{2}^{6}\\right)}{1-2}=378\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-139\">Use the formula to find the indicated partial sum of each geometric series.<\/p>\n\n<div id=\"fs-id1165134037663\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_06\">\n<div id=\"fs-id1165135536527\">\n<p id=\"fs-id1165135536528\">[latex]{S}_{20}[\/latex] for the series[latex]\\text{ 1,000 + 500 + 250 + }\\dots [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135160679\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135160679\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135160679\"]\n<p id=\"fs-id1165135160682\">[latex]\\approx 2,000.00[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-196\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_07\">\n<div id=\"fs-id1165137651760\">\n<p id=\"fs-id1165137651762\">[latex]\\sum _{k=1}^{8}{3}^{k}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137470286\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137470286\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137470286\"]\n<p id=\"fs-id1165137470288\">9,840<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137749990\">\n<div id=\"fs-id1165137749992\">\n<h3>Solving an Application Problem with a Geometric Series<\/h3>\n<p id=\"fs-id1165137749998\">At a new job, an employee\u2019s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137637577\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137637577\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137637577\"]\n<p id=\"fs-id1165137637579\">The problem can be represented by a geometric series with [latex]{a}_{1}=26,750\\text{;}\\,[\/latex][latex]n=5\\text{;}\\,[\/latex]and[latex]\\,r=1.016.[\/latex] Substitute values for[latex]\\,{a}_{1}\\text{,}\\,[\/latex][latex]r\\text{,}[\/latex] and [latex]n[\/latex] into the formula and simplify to find the total amount earned at the end of 5 years.<\/p>\n\n<div id=\"fs-id1165135593139\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{5}=\\frac{26\\text{,}750\\left(1-{1.016}^{5}\\right)}{1-1.016}\\approx 138\\text{,}099.03\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137705448\">He will have earned a total of $138,099.03 by the end of 5 years.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705454\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_08\">\n<div id=\"fs-id1165137476945\">\n<p id=\"fs-id1165137476948\">At a new job, an employee\u2019s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135526997\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135526997\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135526997\"]\n<p id=\"fs-id1165135526999\">$275,513.31<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135527007\" class=\"bc-section section\">\n<h3>Using the Formula for the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165135188378\">Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first [latex]n[\/latex]terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+...[\/latex]<\/p>\n<p id=\"fs-id1165137827407\">This series can also be written in summation notation as [latex]\\sum _{k=1}^{\\infty }2k,[\/latex] where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.<\/p>\n\n<div id=\"fs-id1165135368568\" class=\"bc-section section\">\n<h4>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h4>\n<p id=\"fs-id1165135368573\">If the terms of an <span class=\"no-emphasis\">infinite geometric series<\/span> approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:<\/p>\n\n<div id=\"fs-id1165134181668\" class=\"unnumbered aligncenter\">[latex]1+0.2+0.04+0.008+0.0016+...[\/latex]<\/div>\n<p id=\"fs-id1165134040612\">The common ratio [latex]\\,r\\text{ = 0}\\text{.2}.\\,[\/latex]\nAs[latex]n[\/latex] gets very large, the values of [latex]{r}^{n}[\/latex] get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1&lt;r&lt;1[\/latex] approach 0; the sum of a geometric series is defined when [latex]-1&lt;r&lt;1.[\/latex]<\/p>\n\n<div id=\"fs-id1165137643369\" class=\"textbox key-takeaways\">\n<h3>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h3>\n<p id=\"fs-id1165137643376\">The sum of an infinite series is defined if the series is geometric and [latex]-1&lt;r&lt;1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137932657\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"eip-id1165135606796\"><strong>Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong><\/p>\n\n<ol id=\"fs-id1165137854909\" type=\"1\">\n \t<li>Find the ratio of the second term to the first term.<\/li>\n \t<li>Find the ratio of the third term to the second term.<\/li>\n \t<li>Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.<\/li>\n \t<li>If a common ratio, [latex]r,[\/latex] was found in step 3, check to see if [latex]-1&lt;r&lt;1[\/latex]. If so, the sum is defined. If not, the sum is not defined.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137425843\">\n<div id=\"fs-id1165137425845\">\n<h3>Determining Whether the Sum of an Infinite Series is Defined<\/h3>\n<p id=\"fs-id1165137425850\">Determine whether the sum of each infinite series is defined.<\/p>\n\n<ol id=\"fs-id1165137425853\" type=\"a\">\n \t<li>[latex]\\text{12 + 8 + 4 + }\\dots [\/latex]<\/li>\n \t<li>[latex]\\frac{3}{4}+\\frac{1}{2}+\\frac{1}{3}+...[\/latex]<\/li>\n \t<li>[latex]\\sum _{k=1}^{\\infty }27\\cdot {\\left(\\frac{1}{3}\\right)}^{k}[\/latex]<\/li>\n \t<li>[latex]\\sum _{k=1}^{\\infty }5k[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137843969\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"576145\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"576145\"]\n<ol id=\"fs-id1165137843971\" type=\"a\">\n \t<li>The ratio of the second term to the first is [latex]\\frac{\\text{2}}{\\text{3}},[\/latex]\nwhich is not the same as the ratio of the third term to the second, [latex]\\frac{1}{2}.[\/latex] The series is not geometric.<\/li>\n \t<li>The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of [latex]\\frac{2}{3}\\text{.}[\/latex] The sum of the infinite series is defined.<\/li>\n \t<li>The given formula is exponential with a base of [latex]\\frac{1}{3}\\text{;}[\/latex] the series is geometric with a common ratio of [latex]\\frac{1}{3}\\text{.}[\/latex] The sum of the infinite series is defined.<\/li>\n \t<li>The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-574\">Determine whether the sum of the infinite series is defined.<\/p>\n\n<div id=\"fs-id1165135497153\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137619661\">\n<div id=\"fs-id1165137619663\">\n<p id=\"fs-id1165137619666\">[latex]\\frac{1}{3}+\\frac{1}{2}+\\frac{3}{4}+\\frac{9}{8}+...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137823177\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137823177\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137823177\"]\n<p id=\"fs-id1165137823179\">The sum is not defined.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-827\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_10\">\n<div id=\"fs-id1165137627905\">\n<p id=\"fs-id1165137627907\">[latex]24+\\left(-12\\right)+6+\\left(-3\\right)+...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137400663\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137400663\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137400663\"]\n<p id=\"fs-id1165137400665\">The sum of the infinite series is defined.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-662\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_11\">\n<div id=\"fs-id1165137400674\">\n<p id=\"fs-id1165135159927\">[latex]\\sum _{k=1}^{\\infty }15\\cdot {\\left(\u20130.3\\right)}^{k}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135437872\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135437872\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135437872\"]\n<p id=\"fs-id1165135437874\">The sum of the infinite series is defined.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437882\" class=\"bc-section section\">\n<h4>Finding Sums of Infinite Series<\/h4>\n<p id=\"fs-id1165137679221\">When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first [latex]n[\/latex]terms of a geometric series.<\/p>\n\n<div id=\"fs-id1165135208470\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex]<\/div>\n<p id=\"fs-id1165137552673\">We will examine an infinite series with [latex]r=\\frac{1}{2}.[\/latex] What happens to [latex]{r}^{n}[\/latex] as [latex]n[\/latex] increases?<\/p>\n\n<div id=\"fs-id1165131961708\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(\\frac{1}{2}\\right)}^{2}=\\frac{1}{4}\\\\ {\\left(\\frac{1}{2}\\right)}^{3}=\\frac{1}{8}\\\\ {\\left(\\frac{1}{2}\\right)}^{4}=\\frac{1}{16}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137725370\">The value of[latex]\\,{r}^{n}\\,[\/latex]decreases rapidly. What happens for greater values of [latex]n?[\/latex]<\/p>\n\n<div id=\"fs-id1165135655357\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(\\frac{1}{2}\\right)}^{10}=\\frac{1}{1\\text{,}024}\\hfill \\\\ {\\left(\\frac{1}{2}\\right)}^{20}=\\frac{1}{1\\text{,}048\\text{,}576}\\hfill \\\\ {\\left(\\frac{1}{2}\\right)}^{30}=\\frac{1}{1\\text{,}073\\text{,}741\\text{,}824}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137673611\">As [latex]n[\/latex] gets very large, [latex]{r}^{n}[\/latex] gets very small. We say that, as [latex]n[\/latex] increases without bound, [latex]{r}^{n}[\/latex]approaches 0. As [latex]{r}^{n}[\/latex] approaches 0,[latex]1-{r}^{n}[\/latex] approaches 1. When this happens, the numerator approaches[latex]\\,{a}_{1}.[\/latex] This give us a formula for the sum of an infinite geometric series.<\/p>\n\n<div class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165135187536\">The formula for the sum of an infinite geometric series with [latex]-1&lt;r&lt;1[\/latex] is<\/p>\n\n<div id=\"fs-id1165137851356\">[latex]S=\\frac{{a}_{1}}{1-r}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135190124\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137431516\"><strong>Given an infinite geometric series, find its sum.<\/strong><\/p>\n\n<ol id=\"fs-id1165137431522\" type=\"1\">\n \t<li>Identify[latex]{a}_{1}[\/latex]and [latex]r.[\/latex]<\/li>\n \t<li>Confirm that [latex]\u20131&lt;r&lt;1.[\/latex]<\/li>\n \t<li>Substitute values for [latex]{a}_{1}[\/latex] and [latex]r[\/latex] into the formula, [latex]S=\\frac{{a}_{1}}{1-r}.[\/latex]<\/li>\n \t<li>Simplify to find[latex]\\,S.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137673641\">\n<div id=\"fs-id1165137673643\">\n<h3>Finding the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165137832084\">Find the sum, if it exists, for the following:<\/p>\n\n<ol id=\"fs-id1165137832087\" type=\"a\">\n \t<li>[latex]10+9+8+7+\\dots [\/latex]<\/li>\n \t<li>[latex]248.6+99.44+39.776+\\text{ }\\dots [\/latex]<\/li>\n \t<li>[latex]\\sum _{k=1}^{\\infty }4\\text{,}374\\cdot {\\left(\u2013\\frac{1}{3}\\right)}^{k\u20131}[\/latex]<\/li>\n \t<li>[latex]\\sum _{k=1}^{\\infty }\\frac{1}{9}\\cdot {\\left(\\frac{4}{3}\\right)}^{k}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135394090\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135394090\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135394090\"]\n<ol id=\"fs-id1165135394092\" type=\"a\">\n \t<li>There is not a constant ratio; the series is not geometric.<\/li>\n \t<li>\n<p id=\"fs-id1165135252107\">There is a constant ratio; the series is geometric. [latex]{a}_{1}=248.6[\/latex]and[latex]r=\\frac{99.44}{248.6}=0.4,[\/latex] so the sum exists. Substitute [latex]{a}_{1}=248.6[\/latex] and [latex]r=0.4[\/latex] into the formula and simplify to find the sum:<\/p>\n\n<div id=\"fs-id1165134220772\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}S=\\frac{{a}_{1}}{1-r}\\hfill \\\\ S=\\frac{248.6}{1-0.4}=414.\\overline{3}\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>\n<p id=\"fs-id1165135242798\">The formula is exponential, so the series is geometric with [latex]r=\u2013\\frac{1}{3}.[\/latex] Find[latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula:<\/p>\n\n<div id=\"fs-id1165134284441\" class=\"unnumbered aligncenter\">[latex]{a}_{1}=4\\text{,}374\\cdot {\\left(\u2013\\frac{1}{3}\\right)}^{1\u20131}=4\\text{,}374[\/latex]<\/div>\n<p id=\"fs-id1165135203509\">Substitute [latex]{a}_{1}=4\\text{,}374[\/latex] and [latex]r=-\\frac{1}{3}[\/latex] into the formula, and simplify to find the sum:<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}S=\\frac{{a}_{1}}{1-r}\\hfill \\\\ S=\\frac{4\\text{,}374}{1-\\left(-\\frac{1}{3}\\right)}=3\\text{,}280.5\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>The formula is exponential, so the series is geometric, but[latex]\\,r&gt;1.\\,[\/latex]The sum does not exist.[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137737059\">\n<div id=\"fs-id1165137737061\">\n<h3>Finding an Equivalent Fraction for a Repeating Decimal<\/h3>\n<p id=\"fs-id1165135160513\">Find an equivalent fraction for the repeating decimal [latex]0.\\overline{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135512477\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135512477\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135512477\"]\n<p id=\"fs-id1165135512479\">We notice the repeating decimal [latex]0.\\overline{3}=0.333...[\/latex] so we can rewrite the repeating decimal as a sum of terms.<\/p>\n\n<div id=\"fs-id1165135177536\" class=\"unnumbered aligncenter\">[latex]0.\\overline{3}=0.3+0.03+0.003+...[\/latex]<\/div>\n<p id=\"fs-id1165137761250\">Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.<\/p>\n<span id=\"eip-id1164269283321\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154917\/CNX_Precalc_Figure_11_04_002.jpg\" alt=\"...\"><\/span>\n<p id=\"fs-id1165135195706\">Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have<\/p>\n\n<div id=\"fs-id1165135575202\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}}{1-r}=\\frac{0.3}{1-0.1}=\\frac{0.3}{0.9}=\\frac{1}{3}.[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-120\">Find the sum, if it exists.<\/p>\n\n<div id=\"fs-id1165135208424\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_12\">\n<div id=\"fs-id1165137446138\">\n<p id=\"fs-id1165137446140\">[latex]2+\\frac{2}{3}+\\frac{2}{9}+...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137834920\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834920\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834920\"]\n<p id=\"fs-id1165137834922\">3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1387862\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_13\">\n<div id=\"fs-id1165137834931\">\n<p id=\"fs-id1165137834933\">[latex]\\sum _{k=1}^{\\infty }0.76k+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135596445\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135596445\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135596445\"]\n<p id=\"fs-id1165135596447\">The series is not geometric.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1387940\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_14\">\n<div id=\"fs-id1165134042630\">\n<p id=\"fs-id1165134042632\">[latex]\\sum _{k=1}^{\\infty }{\\left(-\\frac{3}{8}\\right)}^{k}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137387166\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137387166\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137387166\"]\n<p id=\"fs-id1165137387168\">[latex]-\\frac{3}{11}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390954\" class=\"bc-section section\">\n<h3>Solving Annuity Problems<\/h3>\n<p id=\"fs-id1165137794312\">At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% <span class=\"no-emphasis\">annual interest<\/span>, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.<\/p>\n<p id=\"fs-id1165134186203\">We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5%=1.005.[\/latex] After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.<\/p>\n<p id=\"fs-id1165135191421\">We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72.[\/latex] We can substitute [latex]{a}_{1}=50, r=1.005, \\text{and} n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.<\/p>\n\n<div id=\"fs-id1165135531315\" class=\"unnumbered aligncenter\">[latex]{S}_{72}=\\frac{50\\left(1-{1.005}^{72}\\right)}{1-1.005}\\approx 4\\text{,}320.44[\/latex]<\/div>\n<p id=\"fs-id1165135510768\">After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of <strong>72(50) = $3,600<\/strong>.&nbsp; This means that because of the annuity, the couple earned $720.44 interest in their college fund.<\/p>\n\n<div id=\"fs-id1165135510774\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135342192\"><strong>Given an initial deposit and an interest rate, find the value of an annuity.<\/strong><\/p>\n\n<ol type=\"1\">\n \t<li>Determine[latex]\\,{a}_{1}\\text{,}\\,[\/latex]the value of the initial deposit.<\/li>\n \t<li>Determine[latex]\\,n\\text{,}\\,[\/latex]the number of deposits.<\/li>\n \t<li>Determine[latex]\\,r.[\/latex]\n<ol id=\"fs-id1165135187090\" type=\"a\">\n \t<li>Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\n \t<li>Add 1 to this amount to find [latex]r.[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t<li>Substitute values for[latex]\\,{a}_{1}\\text{,}\\,r,\\,\\text{and}\\,n\\,[\/latex]\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series,[latex]{S}_{n}=\\frac{{a}_{1}\\left(1\u2013{r}^{n}\\right)}{1\u2013r}.[\/latex]<\/li>\n \t<li>Simplify to find [latex]{S}_{n},[\/latex] the value of the annuity after [latex]n[\/latex] deposits.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135173286\">\n<div id=\"fs-id1165135173288\">\n<h3>Solving an Annuity Problem<\/h3>\nA deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?\n\n<\/div>\n<div id=\"fs-id1165135388500\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135388500\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135388500\"]\n<p id=\"fs-id1165135388502\">The value of the initial deposit is $100, so[latex]\\,{a}_{1}=100.\\,[\/latex]A total of 120 monthly deposits are made in the 10 years, so[latex]n=120.[\/latex] To find [latex]r,\\,[\/latex]divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.<\/p>\n\n<div class=\"unnumbered\">[latex]r=1+\\frac{0.09}{12}=1.0075[\/latex]<\/div>\n<p id=\"fs-id1165137722693\">Substitute[latex]\\,{a}_{1}=100\\text{,}\\,r=1.0075\\text{,}\\,\\text{and}\\,n=120\\,[\/latex]into the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, and simplify to find the value of the annuity.<\/p>\n\n<div id=\"fs-id1165135582181\" class=\"unnumbered aligncenter\">[latex]{S}_{120}=\\frac{100\\left(1-{1.0075}^{120}\\right)}{1-1.0075}\\approx 19\\text{,}351.43[\/latex]<\/div>\n<p id=\"fs-id1165135503815\">So the account has $19,351.43 after the last deposit is made.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503821\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_15\">\n<div id=\"fs-id1165135503830\">\n<p id=\"fs-id1165137725104\">At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137725110\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137725110\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137725110\"]\n<p id=\"fs-id1165137725113\">$32,775.87<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137725119\" class=\"precalculus media\">\n<p id=\"fs-id1165135188268\">Access these online resources for additional instruction and practice with series.<\/p>\n\n<ul>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/arithmeticser\">Arithmetic Series<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/geometricser\">Geometric Series<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/sumnotation\">Summation Notation<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137740973\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134342462\" summary=\"..\">\n<tbody>\n<tr>\n<td>sum of the first[latex]\\,n\\,[\/latex]\nterms of an arithmetic series<\/td>\n<td>[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of the first[latex]\\,n\\,[\/latex]\nterms of a geometric series<\/td>\n<td>[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r},r\\ne 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of an infinite geometric series with[latex]\\,\u20131&lt;r&lt;\\text{ }1[\/latex]<\/td>\n<td>[latex]{S}_{n}=\\frac{{a}_{1}}{1-r},r\\ne 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137834434\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137834440\">\n \t<li>The sum of the terms in a sequence is called a series.<\/li>\n \t<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See <a class=\"autogenerated-content\" href=\"#Example_11_04_01\">(Figure)<\/a>.<\/li>\n \t<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\n \t<li>The sum of the first[latex]n[\/latex]terms of an arithmetic series can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_04_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_04_03\">(Figure)<\/a>.<\/li>\n \t<li>The sum of the terms in a geometric sequence is called a geometric series.<\/li>\n \t<li>The sum of the first[latex]n[\/latex]terms of a geometric series can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_04_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_04_05\">(Figure)<\/a>.<\/li>\n \t<li>The sum of an infinite series exists if the series is geometric with [latex]\u20131&lt;r&lt;1.[\/latex]<\/li>\n \t<li>If the sum of an infinite series exists, it can be found using a formula. See <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_06\">(Figure)<\/a>, <\/strong><a class=\"autogenerated-content\" href=\"#Example_11_04_07\">(Figure)<\/a><strong>, <\/strong>and <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_08\">(Figure)<\/a><\/strong>.<\/li>\n \t<li>An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_09\">(Figure)<\/a><\/strong>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134109732\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135639904\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135639910\">\n<div id=\"fs-id1165135639912\">\n<p id=\"fs-id1165135639914\">What is an [latex]n\\text{th}[\/latex] partial sum?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134387624\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134387624\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134387624\"]\n<p id=\"fs-id1165134387626\">An [latex]n\\text{th}[\/latex] partial sum is the sum of the first [latex]n[\/latex] terms of a sequence.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635137\">\n<div id=\"fs-id1165137635139\">\n<p id=\"fs-id1165135435776\">What is the difference between an arithmetic sequence and an arithmetic series?<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135435783\">\n<p id=\"fs-id1165135435785\">What is a geometric series?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135435790\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135435790\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135435790\"]\n<p id=\"fs-id1165135435792\">A geometric series is the sum of the terms in a geometric sequence.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693460\">\n<div id=\"fs-id1165137693462\">\n<p id=\"fs-id1165137693464\">How is finding the sum of an infinite geometric series different from finding the [latex]n\\text{th}[\/latex] partial sum?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487089\">\n<div id=\"fs-id1165135487091\">\n<p id=\"fs-id1165135487093\">What is an annuity?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135487097\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135487097\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135487097\"]\n<p id=\"fs-id1165135193165\">An annuity is a series of regular equal payments that earn a constant compounded interest.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193172\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135193177\">For the following exercises, express each description of a sum using summation notation.<\/p>\n\n<div id=\"fs-id1165135193180\">\n<div id=\"fs-id1165135193182\">\n<p id=\"fs-id1165135187266\">The sum of terms [latex]{m}^{2}+3m[\/latex]from [latex]m=1[\/latex] to [latex]m=5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137705183\">\n<p id=\"fs-id1165137705186\">The sum from of [latex]n=0[\/latex] to [latex]n=4[\/latex] of [latex]5n[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135205629\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135205629\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135205629\"]\n<p id=\"fs-id1165135205630\">[latex]\\sum _{n=0}^{4}5n[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137464457\">\n<div id=\"fs-id1165137464459\">\n<p id=\"fs-id1165137464461\">The sum of [latex]6k-5[\/latex] from [latex]k=-2[\/latex] to [latex]k=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333032\">\n<div id=\"fs-id1165135333034\">\n<p id=\"fs-id1165135333036\">The sum that results from adding the number 4 five times<\/p>\n\n<\/div>\n<div id=\"fs-id1165135333040\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135333040\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135333040\"]\n<p id=\"fs-id1165135333042\">[latex]\\sum _{k=1}^{5}4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135168238\">For the following exercises, express each arithmetic sum using summation notation.<\/p>\n\n<div id=\"fs-id1165134149128\">\n<div id=\"fs-id1165134149130\">\n<p id=\"fs-id1165134149132\">[latex]5+10+15+20+25+30+35+40+45+50[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135404219\">\n<div id=\"fs-id1165135190494\">\n<p id=\"fs-id1165135190496\">[latex]10+18+26+\\dots +162[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137696462\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137696462\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137696462\"]\n[latex]\\sum _{k=1}^{20}8k+2[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135194435\">\n<div id=\"fs-id1165135194437\">\n<p id=\"fs-id1165135194440\">[latex]\\frac{1}{2}+1+\\frac{3}{2}+2+\\dots +4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137770272\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each arithmetic sequence.<\/p>\n\n<div>\n<div id=\"fs-id1165137770283\">[latex]\\frac{3}{2}+2+\\frac{5}{2}+3+\\frac{7}{2}[\/latex]<\/div>\n<div id=\"fs-id1165137871515\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871515\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871515\"]\n<p id=\"fs-id1165137871517\">[latex]{S}_{5}=\\frac{5\\left(\\frac{3}{2}+\\frac{7}{2}\\right)}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613235\">\n<div id=\"fs-id1165135613237\">\n<p id=\"fs-id1165135613239\">[latex]19+25+31+\\dots +73[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137745115\">\n<div id=\"fs-id1165137745117\">\n<p id=\"fs-id1165137745119\">[latex]3.2+3.4+3.6+\\dots +5.6[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137619905\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137619905\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137619905\"]\n<p id=\"fs-id1165137834864\">[latex]{S}_{13}=\\frac{13\\left(3.2+5.6\\right)}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135569954\">For the following exercises, express each geometric sum using summation notation.<\/p>\n\n<div id=\"fs-id1165135705038\">\n<div id=\"fs-id1165135705040\">\n<p id=\"fs-id1165135705042\">[latex]1+3+9+27+81+243+729+2187[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137817670\">\n<div id=\"fs-id1165137817672\">[latex]8+4+2+\\dots +0.125[\/latex]<\/div>\n<div id=\"fs-id1165137456659\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137456659\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137456659\"]\n<p id=\"fs-id1165137456662\">[latex]\\sum _{k=1}^{7}8\\cdot {0.5}^{k-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137769967\">\n<div id=\"fs-id1165137769969\">\n<p id=\"fs-id1165137769971\">[latex]-\\frac{1}{6}+\\frac{1}{12}-\\frac{1}{24}+\\dots +\\frac{1}{768}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135203486\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each geometric sequence, and then state the indicated sum.<\/p>\n\n<div id=\"fs-id1165135203490\">\n<div id=\"fs-id1165135203492\">\n<p id=\"fs-id1165135203495\">[latex]9+3+1+\\frac{1}{3}+\\frac{1}{9}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135338212\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135338212\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135338212\"]\n<p id=\"fs-id1165135338214\">[latex]{S}_{5}=\\frac{9\\left(1-{\\left(\\frac{1}{3}\\right)}^{5}\\right)}{1-\\frac{1}{3}}=\\frac{121}{9}\\approx 13.44[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137565170\">\n<div id=\"fs-id1165137565172\">\n<p id=\"fs-id1165137565174\">[latex]\\sum _{n=1}^{9}5\\cdot {2}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503927\">\n<div id=\"fs-id1165135503929\">\n<p id=\"fs-id1165135503931\">[latex]\\sum _{a=1}^{11}64\\cdot {0.2}^{a-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137704468\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137704468\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137704468\"]\n<p id=\"fs-id1165135205888\">[latex]{S}_{11}=\\frac{64\\left(1-{0.2}^{11}\\right)}{1-0.2}=\\frac{781,249,984}{9,765,625}\\approx 80[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135417823\">For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.<\/p>\n\n<div id=\"fs-id1165135417827\">\n<div id=\"fs-id1165137741039\">\n<p id=\"fs-id1165137741041\">[latex]12+18+24+30+...[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137862580\">[latex]2+1.6+1.28+1.024+...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135496306\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135496306\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135496306\"]\n<p id=\"fs-id1165135496308\">The series is defined. [latex]S=\\frac{2}{1-0.8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333611\">\n<div id=\"fs-id1165135333613\">\n<p id=\"fs-id1165135333615\">[latex]\\sum _{m=1}^{\\infty }{4}^{m-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571886\">\n<div id=\"fs-id1165135571888\">[latex]\\underset{\\infty }{\\overset{k=1}{{\\sum }^{\\text{\u200b}}}}-{\\left(-\\frac{1}{2}\\right)}^{k-1}[\/latex]<\/div>\n<div id=\"fs-id1165137701119\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137701119\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137701119\"]\n<p id=\"fs-id1165135408472\">The series is defined. [latex]S=\\frac{-1}{1-\\left(-\\frac{1}{2}\\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827903\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137894472\">For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.<\/p>\n\n<div id=\"fs-id1165137894476\">\n<div id=\"fs-id1165137894478\">\n\nGraph the arithmetic sequence showing one year of Javier\u2019s deposits.\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137894486\">\n<div id=\"fs-id1165137894488\">\n<p id=\"fs-id1165137828073\">Graph the arithmetic series showing the monthly sums of one year of Javier\u2019s deposits.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137828079\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137828079\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137828079\"]<span id=\"fs-id1165137828084\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154923\/CNX_Precalc_Figure_11_04_202.jpg\" alt=\"Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165137827820\">For the following exercises, use the geometric series[latex]{\\sum _{k=1}^{\\infty }\\left(\\frac{1}{2}\\right)}^{k}.[\/latex]<\/p>\n\n<div id=\"fs-id1165137761311\">\n<div id=\"fs-id1165137761314\">\n<p id=\"fs-id1165137761316\">Graph the first 7 partial sums of the series.<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137642999\">\n<p id=\"fs-id1165137643001\">What number does [latex]{S}_{n}[\/latex] seem to be approaching in the graph? Find the sum to explain why this makes sense.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135208860\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135208860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135208860\"]\n<p id=\"fs-id1165135208862\">Sample answer: The graph of [latex]{S}_{n}[\/latex] seems to be approaching 1. This makes sense because[latex]\\sum _{k=1}^{\\infty }{\\left(\\frac{1}{2}\\right)}^{k}[\/latex]is a defined infinite geometric series with [latex]S=\\frac{\\frac{1}{2}}{1\u2013\\left(\\frac{1}{2}\\right)}=1.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203672\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135701517\">For the following exercises, find the indicated sum.<\/p>\n\n<div id=\"fs-id1165135701521\">\n<div id=\"fs-id1165135701523\">\n<p id=\"fs-id1165135701525\">[latex]\\sum _{a=1}^{14}a[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891293\">\n<div id=\"fs-id1165137891295\">\n<p id=\"fs-id1165137891297\">[latex]\\sum _{n=1}^{6}n\\left(n-2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137453619\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137453619\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137453619\"]\n<p id=\"fs-id1165137453621\">49<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453626\">\n<div id=\"fs-id1165137453629\">[latex]\\sum _{k=1}^{17}{k}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137736016\">\n<div id=\"fs-id1165137736018\">\n<p id=\"fs-id1165137736020\">[latex]\\sum _{k=1}^{7}{2}^{k}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134183115\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134183115\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134183115\"]\n<p id=\"fs-id1165134183117\">254<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134183122\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series to find the sum.<\/p>\n\n<div id=\"fs-id1165135194113\">\n<div id=\"fs-id1165135194116\">\n<p id=\"fs-id1165135194118\">[latex]-1.7+-0.4+0.9+2.2+3.5+4.8[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723368\">\n<div id=\"fs-id1165137807476\">\n<p id=\"fs-id1165137807478\">[latex]6+\\frac{15}{2}+9+\\frac{21}{2}+12+\\frac{27}{2}+15[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135471078\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135471078\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135471078\"]\n<p id=\"fs-id1165135471080\">[latex]{S}_{7}=\\frac{147}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042711\">\n<div id=\"fs-id1165134042713\">\n<p id=\"fs-id1165137673868\">[latex]-1+3+7+...+31[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849073\">\n<div id=\"fs-id1165137849075\">[latex]\\sum _{k=1}^{11}\\left(\\frac{k}{2}-\\frac{1}{2}\\right)[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"656521\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"656521\"][latex]{S}_{11}=\\frac{55}{2}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135301093\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of a geometric series to find the partial sum.<\/p>\n\n<div id=\"fs-id1165135516941\">\n<div id=\"fs-id1165135516943\">\n\n[latex]{S}_{6}[\/latex] for the series [latex]-2-10-50-250...[\/latex]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137784646\">[latex]{S}_{7}[\/latex] for the series [latex]0.4-2+10-50...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137530296\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137530296\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137530296\"]\n<p id=\"fs-id1165137530298\">[latex]{S}_{7}=5208.4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165135187164\">[latex]\\sum _{k=1}^{9}{2}^{k-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134094671\">\n<div id=\"fs-id1165134094673\">\n<p id=\"fs-id1165134094676\">[latex]\\sum _{n=1}^{10}-2\\cdot {\\left(\\frac{1}{2}\\right)}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135496397\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135496397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135496397\"]\n<p id=\"fs-id1165135496400\">[latex]{S}_{10}=-\\frac{1023}{256}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137749444\">For the following exercises, find the sum of the infinite geometric series.<\/p>\n\n<div id=\"fs-id1165137749447\">\n<div id=\"fs-id1165137749449\">\n<p id=\"fs-id1165137749451\">[latex]4+2+1+\\frac{1}{2}...[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137738012\">\n<div id=\"fs-id1165137738014\">\n<p id=\"fs-id1165137761900\">[latex]-1-\\frac{1}{4}-\\frac{1}{16}-\\frac{1}{64}...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137535603\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137535603\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137535603\"]\n<p id=\"fs-id1165137535605\">[latex]S=-\\frac{4}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135332680\">\n<div id=\"fs-id1165135332682\">\n<p id=\"fs-id1165135332684\">[latex]\\underset{\\infty }{\\overset{k=1}{{\\sum }^{\\text{\u200b}}}}3\\cdot {\\left(\\frac{1}{4}\\right)}^{k-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734577\">\n<div id=\"fs-id1165137734579\">\n<p id=\"fs-id1165135342830\">[latex]\\sum _{n=1}^{\\infty }4.6\\cdot {0.5}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135333829\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135333829\"]\n<p id=\"fs-id1165135333829\">[latex]S=9.2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137556920\">For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.<\/p>\n\n<div id=\"fs-id1165137556925\">\n<div>\n<p id=\"fs-id1165137556929\">Deposit amount: [latex]\\text{\\$}50;[\/latex] total deposits: [latex]60;[\/latex] interest rate: [latex]5%,[\/latex] compounded monthly<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205791\">\n<div id=\"fs-id1165135205793\">\n<p id=\"fs-id1165135205796\">Deposit amount: [latex]\\text{\\$}150;[\/latex] total deposits: [latex]24;[\/latex] interest rate: [latex]3%,[\/latex] compounded monthly<\/p>\n\n<\/div>\n<div id=\"fs-id1165137705498\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137705498\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137705498\"]\n<p id=\"fs-id1165137705500\">$3,705.42<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705506\">\n<div id=\"fs-id1165137705508\">\n<p id=\"fs-id1165137705510\">Deposit amount: [latex]\\text{\\$}450;[\/latex] total deposits: [latex]60;[\/latex] interest rate: [latex]4.5%,[\/latex] compounded quarterly<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134157171\">\n<div id=\"fs-id1165134157173\">\n<p id=\"fs-id1165134157175\">Deposit amount: [latex]\\text{\\$}100;[\/latex] total deposits: [latex]120;[\/latex] interest rate: [latex]10%,[\/latex] compounded semi-annually<\/p>\n\n<\/div>\n<div id=\"fs-id1165135481987\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135481987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135481987\"]\n<p id=\"fs-id1165135481990\">$695,823.97<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135481996\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135482001\">\n<div id=\"fs-id1165135422912\">\n<p id=\"fs-id1165135422915\">The sum of terms [latex]50-{k}^{2}[\/latex] from [latex]k=x[\/latex] through [latex]7[\/latex] is [latex]115.[\/latex] What is <em>x<\/em>?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188088\">\n<div id=\"fs-id1165135188090\">\n<p id=\"fs-id1165135188092\">Write an explicit formula for[latex]{a}_{k}[\/latex]such that[latex]\\sum _{k=0}^{6}{a}_{k}=189.[\/latex] Assume this is an arithmetic series.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135245742\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135245742\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135245742\"]\n<p id=\"fs-id1165135245744\">[latex]{a}_{k}=30-k[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134212315\">\n<div id=\"fs-id1165134212317\">\n<p id=\"fs-id1165134212319\">Find the smallest value of <em>n<\/em> such that[latex]\\sum _{k=1}^{n}\\left(3k\u20135\\right)&gt;100.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135500068\">\n<div id=\"fs-id1165135500070\">\n<p id=\"fs-id1165135500072\">How many terms must be added before the series [latex]-1-3-5-7....\\text{ }[\/latex]has a sum less than [latex]-75?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134061152\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134061152\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134061152\"]\n<p id=\"fs-id1165134061154\">9 terms<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134061159\">\n<div id=\"fs-id1165134061161\">\n<p id=\"fs-id1165134061163\">Write [latex]0.\\overline{65}[\/latex] as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert [latex]0.\\overline{65}[\/latex] to a fraction.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134406874\">\n<div id=\"fs-id1165134406876\">\n<p id=\"fs-id1165137817620\">The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137817626\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137817626\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137817626\"]\n<p id=\"fs-id1165137817628\">[latex]r=\\frac{4}{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191488\">\n<div id=\"fs-id1165135191490\">\n<p id=\"fs-id1165135191492\">To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191500\">\n<div id=\"fs-id1165135191503\">\n<p id=\"fs-id1165135191505\">Karl has two years to save [latex]$10,000[\/latex] to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135181468\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181468\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181468\"]\n<p id=\"fs-id1165135181470\">$400 per month<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187471\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135187477\">\n<div id=\"fs-id1165135187479\">\n<p id=\"fs-id1165135187481\">Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for[latex]\\,1\\,[\/latex]hour, and each successive day she will increase her study time by[latex]\\,30\\,[\/latex]minutes. How many hours will Keisha have studied after one week?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181425\">\n<div id=\"fs-id1165135181427\">\n<p id=\"fs-id1165135181429\">A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135181435\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181435\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181435\"]\n<p id=\"fs-id1165135181437\">420 feet<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181442\">\n<div id=\"fs-id1165135181445\">\n<p id=\"fs-id1165137696121\">A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696127\">\n<div id=\"fs-id1165137696130\">\n<p id=\"fs-id1165137696132\">A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels [latex]\\frac{3}{4}[\/latex] the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137629057\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137629057\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137629057\"]\n<p id=\"fs-id1165137629059\">12 feet<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137629064\">\n<div id=\"fs-id1165137629067\">\n<p id=\"fs-id1165137629069\">Rachael deposits $1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137726792\">\n \t<dt>annuity<\/dt>\n \t<dd id=\"fs-id1165137726797\">an investment in which the purchaser makes a sequence of periodic, equal payments<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137726801\">\n \t<dt>arithmetic series<\/dt>\n \t<dd id=\"fs-id1165137726806\">the sum of the terms in an arithmetic sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137726811\">\n \t<dt>diverge<\/dt>\n \t<dd>a series is said to diverge if the sum is not a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134031386\">\n \t<dt>geometric series<\/dt>\n \t<dd id=\"fs-id1165134031391\">the sum of the terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134031396\">\n \t<dt>index of summation<\/dt>\n \t<dd id=\"fs-id1165134031401\">in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation<\/dd>\n<\/dl>\n<dl>\n \t<dt>infinite series<\/dt>\n \t<dd id=\"fs-id1165137737867\">the sum of the terms in an infinite sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137737872\">\n \t<dt>lower limit of summation<\/dt>\n \t<dd id=\"fs-id1165137737877\">the number used in the explicit formula to find the first term in a series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137737881\">\n \t<dt>nth partial sum<\/dt>\n \t<dd id=\"fs-id1165135471113\">the sum of the first[latex]n[\/latex]terms of a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135471124\">\n \t<dt>series<\/dt>\n \t<dd id=\"fs-id1165135471129\">the sum of the terms in a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135471133\">\n \t<dt>summation notation<\/dt>\n \t<dd id=\"fs-id1165135471138\">a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762744\">\n \t<dt>upper limit of summation<\/dt>\n \t<dd id=\"fs-id1165137762749\">the number used in the explicit formula to find the last term in a series<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use summation notation.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\n<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\n<li>Solve annuity problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137612041\">A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.<\/p>\n<div id=\"fs-id1165137405637\" class=\"bc-section section\">\n<h3>Using Summation Notation<\/h3>\n<p id=\"fs-id1165137788981\">To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.<\/p>\n<div id=\"fs-id1165135317537\" class=\"unnumbered aligncenter\">[latex]3+7+11+15+19+...[\/latex]<\/div>\n<p id=\"fs-id1165137639252\">The [latex]n\\text{th }[\/latex]partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation[latex]\\text{ }{S}_{n}\\text{ }[\/latex]represents the partial sum.<\/p>\n<div id=\"eip-901\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{1}=3\\\\ {S}_{2}=3+7=10\\\\ {S}_{3}=3+7+11=21\\\\ {S}_{4}=3+7+11+15=36\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137653088\"><strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <span class=\"no-emphasis\">sigma<\/span>, [latex]\\text{\u03a3},[\/latex] to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.<\/p>\n<p><span id=\"fs-id1165137432096\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154910\/CNX_Precalc_Figure_11_04_001n.jpg\" alt=\"Explanation of summation notion as described in the text.\" \/><\/span><\/p>\n<p id=\"fs-id1165137758894\">If we interpret the given notation, we see that it asks us to find the sum of the terms in the series[latex]\\,{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5.\\,[\/latex] We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.<\/p>\n<div id=\"fs-id1165137726334\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{1}=2\\left(1\\right)=2\\end{array}\\hfill \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137678008\">We can find the sum of the series by adding the terms:<\/p>\n<div id=\"fs-id1165135481276\" class=\"unnumbered aligncenter\">[latex]\\sum _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/div>\n<div id=\"fs-id1165137726612\" class=\"textbox key-takeaways\">\n<h3>Summation Notation<\/h3>\n<p id=\"fs-id1165137471824\">The sum of the first[latex]n[\/latex]terms of a <strong>series <\/strong>can be expressed in summation notation as follows:<\/p>\n<div id=\"fs-id1165137938486\" class=\"unnumbered aligncenter\">[latex]\\sum _{k=1}^{n}{a}_{k}[\/latex]<\/div>\n<p id=\"fs-id1165134109690\">This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from [latex]k=1[\/latex] to [latex]k=n.[\/latex]<\/p>\n<p id=\"eip-419\">[latex]k\\,[\/latex] is called the index of summation, 1 is the lower limit of summation, and [latex]n[\/latex] is the upper limit of summation.<\/p>\n<\/div>\n<div id=\"fs-id1165137732268\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1165135496558\"><strong>Does the lower limit of summation have to be 1?<\/strong><\/p>\n<p id=\"fs-id1165137727904\"><em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em><\/p>\n<\/div>\n<div class=\"precalculus howto\">\n<p id=\"fs-id1165137761194\"><strong>Given summation notation for a series, evaluate the value.<\/strong><\/p>\n<ol id=\"fs-id1165137731973\" type=\"1\">\n<li>Identify the lower limit of summation.<\/li>\n<li>Identify the upper limit of summation.<\/li>\n<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\n<li>Add to find the sum.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1165135209801\">\n<div id=\"fs-id1165137466005\">\n<h3>Using Summation Notation<\/h3>\n<p id=\"fs-id1165137834859\">Evaluate[latex]\\sum _{k=3}^{7}{k}^{2}.[\/latex]<\/p>\n<\/div>\n<div>\n<div id=\"fs-id1165137464424\" class=\"unnumbered aligncenter\">\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137472839\">According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7.[\/latex] We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6\\text{,}\\,[\/latex] and [latex]7[\/latex] into the function [latex]{k}^{2}.[\/latex] We add the terms to find the sum.<\/p>\n<div id=\"fs-id1165137464424\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\sum _{k=3}^{7}{k}^{2}\\hfill & ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill & =9+16+25+36+49\\hfill \\\\ \\hfill & =135\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482825\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137702091\">\n<p id=\"fs-id1165137806341\">Evaluate[latex]\\sum _{k=2}^{5}\\left(3k\u20131\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137658521\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137589299\">38<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137480338\" class=\"bc-section section\">\n<h3>Using the Formula for Arithmetic Series<\/h3>\n<p id=\"fs-id1165137557625\">Just as we studied special types of sequences, we will look at special types of series. Recall that an <span class=\"no-emphasis\">arithmetic sequence<\/span> is a sequence in which the difference between any two consecutive terms is the <span class=\"no-emphasis\">common difference<\/span>,[latex]d.[\/latex] The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:<\/p>\n<div id=\"eip-id1165137653029\" class=\"unnumbered\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}\u2013d\\right)+{a}_{n}.[\/latex]<\/div>\n<p id=\"fs-id1165135176760\">We can also reverse the order of the terms and write the sum as<\/p>\n<div id=\"eip-id1165137874798\" class=\"unnumbered\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}\u2013d\\right)+\\left({a}_{n}\u20132d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}.[\/latex]<\/div>\n<p>If we add these two expressions for the sum of the first [latex]n[\/latex]terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.<\/p>\n<div id=\"eip-id1165135169564\" class=\"unnumbered\">[latex]\\frac{\\begin{array}{l}\\,\\,\\,\\,\\,\\,{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}\u2013d\\right)+{a}_{n}\\hfill \\\\ +\\,\\,{S}_{n}={a}_{n}+\\left({a}_{n}\u2013d\\right)+\\left({a}_{n}\u20132d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}\\hfill \\end{array}}{2{S}_{n}=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165134148512\">Because there are [latex]n[\/latex] terms in the series, we can simplify this sum to<\/p>\n<div id=\"eip-id1165137726382\" class=\"unnumbered\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right).[\/latex]<\/div>\n<p id=\"fs-id1165135505002\">We divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.<\/p>\n<div id=\"eip-id1165137447374\" class=\"unnumbered\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/div>\n<div id=\"fs-id1165135506407\" class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of the First <em>n<\/em> Terms of an Arithmetic Series<\/h3>\n<p id=\"fs-id1165134040601\">An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first [latex]n[\/latex] terms of an arithmetic sequence is<\/p>\n<div id=\"fs-id1165137741027\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135255874\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137529710\"><strong>Given terms of an arithmetic series, find the sum of the first [latex]n[\/latex] terms.<\/strong><\/p>\n<ol id=\"fs-id1165135646144\" type=\"1\">\n<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}.[\/latex]<\/li>\n<li>Determine [latex]n.[\/latex]<\/li>\n<li>Substitute values for [latex]{a}_{1}\\text{, }{a}_{n}\\text{,}\\,[\/latex] and [latex]\\,n\\,[\/latex] into the formula [latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}.[\/latex]<\/li>\n<li>Simplify to find [latex]{S}_{n}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135191078\">\n<div id=\"fs-id1165135191081\">\n<h3>Finding the First <em>n<\/em> Terms of an Arithmetic Series<\/h3>\n<p id=\"fs-id1165135159870\">Find the sum of each arithmetic series.<\/p>\n<ol id=\"fs-id1165135159873\" type=\"a\">\n<li>[latex]\\text{5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32}[\/latex]<\/li>\n<li>[latex]\\text{20 + 15 + 10 +\u2026+ \u221250}[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{12}3k-8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135571638\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135571640\" type=\"a\">\n<li>\n<p id=\"fs-id1165137665556\">We are given [latex]{a}_{1}=5[\/latex] and [latex]\\,{a}_{n}=32.[\/latex]<\/p>\n<p id=\"fs-id1165135404253\">Count the number of terms in the sequence to find [latex]n=10.[\/latex]<\/p>\n<p id=\"fs-id1165135417695\">Substitute values for [latex]\\,{a}_{1},{a}_{n}\\text{\\hspace{0.17em},}[\/latex] and [latex]n[\/latex] into the formula and simplify.<\/p>\n<div id=\"eip-id1165137748560\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\end{array}\\hfill \\\\ {S}_{10}=\\frac{10\\left(5+32\\right)}{2}=185\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165137832192\">We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50.[\/latex]<\/p>\n<p id=\"fs-id1165135186923\">Use the formula for the general term of an arithmetic sequence to find [latex]n.[\/latex]<\/p>\n<div id=\"eip-id1165137806527\" class=\"unnumbered\">[latex]\\begin{array}{l}\\,\\,\\,\\,{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ -50=20+\\left(n-1\\right)\\left(-5\\right)\\hfill \\\\ -70=\\left(n-1\\right)\\left(-5\\right)\\hfill \\\\ \\,\\,\\,\\,14=n-1\\hfill \\\\ \\,\\,\\,\\,15=n\\hfill \\end{array}[\/latex]<\/div>\n<p>Substitute values for [latex]{a}_{1},{a}_{n}\\text{,}\\,n[\/latex] into the formula and simplify.<\/p>\n<div><\/div>\n<div id=\"eip-389\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\end{array}\\hfill \\\\ {S}_{15}=\\frac{15\\left(20-50\\right)}{2}=-225\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165135175094\">To find [latex]{a}_{1},\\,[\/latex]substitute [latex]k=1[\/latex] into the given explicit formula.<\/p>\n<div id=\"eip-id1165135528406\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{k}=3k-8\\hfill \\\\ \\text{ }{a}_{1}=3\\left(1\\right)-8=-5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137627558\">We are given that [latex]n=12.[\/latex] To find [latex]{a}_{12},\\,[\/latex]substitute [latex]k=12[\/latex] into the given explicit formula.<\/p>\n<div id=\"eip-532\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }{a}_{k}=3k-8\\hfill \\\\ {a}_{12}=3\\left(12\\right)-8=28\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137641682\">Substitute values for [latex]{a}_{1},{a}_{n},[\/latex] and [latex]n[\/latex] into the formula and simplify.<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ {S}_{12}=\\frac{12\\left(-5+28\\right)}{2}=138\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-32\">Use the formula to find the sum of each arithmetic series.<\/p>\n<div id=\"fs-id1165134032286\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_02\">\n<div id=\"fs-id1165134092405\">\n<p id=\"eip-id1926809\">[latex]\\text{1}\\text{.4 + 1}\\text{.6 + 1}\\text{.8 + 2}\\text{.0 + 2}\\text{.2 + 2}\\text{.4 + 2}\\text{.6 + 2}\\text{.8 + 3}\\text{.0 + 3}\\text{.2 + 3}\\text{.4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134226148\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134226150\">[latex]\\text{26}\\text{.4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_03\">\n<div id=\"fs-id1165135198570\">\n<p id=\"eip-id1862925\">[latex]\\text{13 + 21 + 29 + }\\dots \\text{+ 69}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135198577\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135198579\">[latex]\\text{328}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_04\">\n<div id=\"fs-id1165137651940\">\n<p id=\"eip-id2303118\">[latex]\\sum _{k=1}^{10}5-6k[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137874274\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137874276\">[latex]\\text{\u2212280}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135208460\">\n<div id=\"fs-id1165135208462\">\n<h3>Solving Application Problems with Arithmetic Series<\/h3>\n<p id=\"fs-id1165134382167\">On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?<\/p>\n<\/div>\n<div id=\"fs-id1165134382173\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134382175\">This problem can be modeled by an arithmetic series with[latex]\\,{a}_{1}=\\frac{1}{2}\\,[\/latex]and[latex]\\,d=\\frac{1}{4}.\\,[\/latex]We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8\\text{,}[\/latex] and we are looking for[latex]\\,{S}_{8}.\\,[\/latex]To find [latex]{a}_{8},[\/latex] we can use the explicit formula for an arithmetic sequence.<\/p>\n<div id=\"eip-id1165135512530\" class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{n}={a}_{1}+d\\left(n-1\\right)\\end{array}\\hfill \\\\ {a}_{8}=\\frac{1}{2}+\\frac{1}{4}\\left(8-1\\right)=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137771149\">We can now use the formula for arithmetic series.<\/p>\n<div id=\"eip-id1165137653366\" class=\"unnumbered\">[latex]\\begin{array}{l} {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ \\text{ }{S}_{8}=\\frac{8\\left(\\frac{1}{2}+\\frac{9}{4}\\right)}{2}=11\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137673659\">She will have walked a total of 11 miles.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673666\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_05\">\n<div id=\"fs-id1165137734534\">\n<p id=\"fs-id1165137734536\">A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?<\/p>\n<\/div>\n<div id=\"fs-id1165137660095\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137660097\">$2,025<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660105\" class=\"bc-section section\">\n<h3>Using the Formula for Geometric Series<\/h3>\n<p id=\"fs-id1165135188612\">Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a <strong>geometric series<\/strong>. Recall that a <span class=\"no-emphasis\">geometric sequence<\/span> is a sequence in which the ratio of any two consecutive terms is the <span class=\"no-emphasis\">common ratio<\/span>, [latex]\\,r.\\,[\/latex]We can write the sum of the first [latex]n[\/latex] terms of a geometric series as<\/p>\n<div class=\"unnumbered\">[latex]{S}_{n}={a}_{1}+r{a}_{1}+{r}^{2}{a}_{1}+...+{r}^{n\u20131}{a}_{1}.[\/latex]<\/div>\n<p id=\"fs-id1165135504988\">Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first[latex]\\,n\\,[\/latex]terms of a geometric series. We will begin by multiplying both sides of the equation by[latex]\\,r.\\,[\/latex]<\/p>\n<div id=\"eip-41\" class=\"unnumbered aligncenter\">[latex]r{S}_{n}=r{a}_{1}+{r}^{2}{a}_{1}+{r}^{3}{a}_{1}+...+{r}^{n}{a}_{1}[\/latex]<\/div>\n<p id=\"fs-id1165134172597\">Next, we subtract this equation from the original equation.<\/p>\n<div id=\"fs-id1165135704813\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\frac{\\begin{array}{l}\\text{ }{S}_{n}={a}_{1}+r{a}_{1}+{r}^{2}{a}_{1}+...+{r}^{n\u20131}{a}_{1}\\hfill \\\\ -r{S}_{n}=-\\left(r{a}_{1}+{r}^{2}{a}_{1}+{r}^{3}{a}_{1}+...+{r}^{n}{a}_{1}\\right)\\hfill \\end{array}}{\\left(1-r\\right){S}_{n}={a}_{1}-{r}^{n}{a}_{1}}\\end{array}[\/latex]<\/div>\n<p>Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n},[\/latex] divide both sides by [latex]\\left(1-r\\right).[\/latex]<\/p>\n<div id=\"fs-id1165137871953\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/div>\n<div id=\"fs-id1165137533230\" class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of the First <em>n<\/em> Terms of a Geometric Series<\/h3>\n<p id=\"fs-id1165137863800\">A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first[latex]\\,n\\,[\/latex]terms of a geometric sequence is represented as<\/p>\n<div>[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137602804\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p><strong>Given a geometric series, find the sum of the first <em>n<\/em> terms.<\/strong><\/p>\n<ol id=\"fs-id1165137698300\" type=\"1\">\n<li>Identify[latex]\\,{a}_{1},\\,r,\\,\\text{and}\\,n.[\/latex]<\/li>\n<li>Substitute values for[latex]\\,{a}_{1},\\,r,[\/latex] and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\frac{{a}_{1}\\left(1\u2013{r}^{n}\\right)}{1\u2013r}.[\/latex]<\/li>\n<li>Simplify to find [latex]{S}_{n}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135196954\">\n<div id=\"fs-id1165135196957\">\n<h3>Finding the First <em>n<\/em> Terms of a Geometric Series<\/h3>\n<p>Use the formula to find the indicated partial sum of each geometric series.<\/p>\n<ol id=\"fs-id1165137736582\" type=\"a\">\n<li>[latex]{S}_{11}[\/latex]for the series[latex]\\text{ 8 + -4 + 2 + }\\dots[\/latex]<\/li>\n<li>[latex]\\underset{k=1}{\\overset{6}{{\\sum }^{\\text{\u200b}}}}3\\cdot {2}^{k}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135264856\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135264859\" type=\"a\">\n<li>\n<p id=\"fs-id1165137732247\">[latex]{a}_{1}=8,[\/latex] and we are given that [latex]n=11.[\/latex]<\/p>\n<p id=\"fs-id1165135255878\">We can find [latex]r[\/latex] by dividing the second term of the series by the first.<\/p>\n<div id=\"fs-id1165135564192\" class=\"unnumbered aligncenter\">[latex]r=\\frac{-4}{8}=-\\frac{1}{2}[\/latex]<\/div>\n<p id=\"fs-id1165137755996\">Substitute values for [latex]{a}_{1}, r, \\text{and} n[\/latex] into the formula and simplify.<\/p>\n<div id=\"fs-id1165133347581\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{11}=\\frac{8\\left(1-{\\left(-\\frac{1}{2}\\right)}^{11}\\right)}{1-\\left(-\\frac{1}{2}\\right)}\\approx 5.336\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165137673547\">Find [latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.<\/p>\n<div id=\"fs-id1165137932596\" class=\"unnumbered aligncenter\">[latex]{a}_{1}=3\\cdot {2}^{1}=6[\/latex]<\/div>\n<p id=\"fs-id1165137656631\">We can see from the given explicit formula that [latex]r=2.[\/latex] The upper limit of summation is 6, so [latex]n=6.[\/latex]<\/p>\n<p id=\"fs-id1165135149030\">Substitute values for [latex]{a}_{1},\\,r,[\/latex] and [latex]n[\/latex] into the formula, and simplify.<\/p>\n<div id=\"fs-id1165137400638\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{6}=\\frac{6\\left(1-{2}^{6}\\right)}{1-2}=378\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-139\">Use the formula to find the indicated partial sum of each geometric series.<\/p>\n<div id=\"fs-id1165134037663\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_06\">\n<div id=\"fs-id1165135536527\">\n<p id=\"fs-id1165135536528\">[latex]{S}_{20}[\/latex] for the series[latex]\\text{ 1,000 + 500 + 250 + }\\dots[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135160679\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135160682\">[latex]\\approx 2,000.00[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-196\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_07\">\n<div id=\"fs-id1165137651760\">\n<p id=\"fs-id1165137651762\">[latex]\\sum _{k=1}^{8}{3}^{k}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137470286\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137470288\">9,840<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137749990\">\n<div id=\"fs-id1165137749992\">\n<h3>Solving an Application Problem with a Geometric Series<\/h3>\n<p id=\"fs-id1165137749998\">At a new job, an employee\u2019s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.<\/p>\n<\/div>\n<div id=\"fs-id1165137637577\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137637579\">The problem can be represented by a geometric series with [latex]{a}_{1}=26,750\\text{;}\\,[\/latex][latex]n=5\\text{;}\\,[\/latex]and[latex]\\,r=1.016.[\/latex] Substitute values for[latex]\\,{a}_{1}\\text{,}\\,[\/latex][latex]r\\text{,}[\/latex] and [latex]n[\/latex] into the formula and simplify to find the total amount earned at the end of 5 years.<\/p>\n<div id=\"fs-id1165135593139\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\hfill \\\\ {S}_{5}=\\frac{26\\text{,}750\\left(1-{1.016}^{5}\\right)}{1-1.016}\\approx 138\\text{,}099.03\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137705448\">He will have earned a total of $138,099.03 by the end of 5 years.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705454\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_08\">\n<div id=\"fs-id1165137476945\">\n<p id=\"fs-id1165137476948\">At a new job, an employee\u2019s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?<\/p>\n<\/div>\n<div id=\"fs-id1165135526997\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135526999\">$275,513.31<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135527007\" class=\"bc-section section\">\n<h3>Using the Formula for the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165135188378\">Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first [latex]n[\/latex]terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+...[\/latex]<\/p>\n<p id=\"fs-id1165137827407\">This series can also be written in summation notation as [latex]\\sum _{k=1}^{\\infty }2k,[\/latex] where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.<\/p>\n<div id=\"fs-id1165135368568\" class=\"bc-section section\">\n<h4>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h4>\n<p id=\"fs-id1165135368573\">If the terms of an <span class=\"no-emphasis\">infinite geometric series<\/span> approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:<\/p>\n<div id=\"fs-id1165134181668\" class=\"unnumbered aligncenter\">[latex]1+0.2+0.04+0.008+0.0016+...[\/latex]<\/div>\n<p id=\"fs-id1165134040612\">The common ratio [latex]\\,r\\text{ = 0}\\text{.2}.\\,[\/latex]<br \/>\nAs[latex]n[\/latex] gets very large, the values of [latex]{r}^{n}[\/latex] get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1<r<1[\/latex] approach 0; the sum of a geometric series is defined when [latex]-1<r<1.[\/latex]<\/p>\n<div id=\"fs-id1165137643369\" class=\"textbox key-takeaways\">\n<h3>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h3>\n<p id=\"fs-id1165137643376\">The sum of an infinite series is defined if the series is geometric and [latex]-1<r<1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137932657\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"eip-id1165135606796\"><strong>Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong><\/p>\n<ol id=\"fs-id1165137854909\" type=\"1\">\n<li>Find the ratio of the second term to the first term.<\/li>\n<li>Find the ratio of the third term to the second term.<\/li>\n<li>Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.<\/li>\n<li>If a common ratio, [latex]r,[\/latex] was found in step 3, check to see if [latex]-1<r<1[\/latex]. If so, the sum is defined. If not, the sum is not defined.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137425843\">\n<div id=\"fs-id1165137425845\">\n<h3>Determining Whether the Sum of an Infinite Series is Defined<\/h3>\n<p id=\"fs-id1165137425850\">Determine whether the sum of each infinite series is defined.<\/p>\n<ol id=\"fs-id1165137425853\" type=\"a\">\n<li>[latex]\\text{12 + 8 + 4 + }\\dots[\/latex]<\/li>\n<li>[latex]\\frac{3}{4}+\\frac{1}{2}+\\frac{1}{3}+...[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{\\infty }27\\cdot {\\left(\\frac{1}{3}\\right)}^{k}[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{\\infty }5k[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137843969\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137843971\" type=\"a\">\n<li>The ratio of the second term to the first is [latex]\\frac{\\text{2}}{\\text{3}},[\/latex]<br \/>\nwhich is not the same as the ratio of the third term to the second, [latex]\\frac{1}{2}.[\/latex] The series is not geometric.<\/li>\n<li>The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of [latex]\\frac{2}{3}\\text{.}[\/latex] The sum of the infinite series is defined.<\/li>\n<li>The given formula is exponential with a base of [latex]\\frac{1}{3}\\text{;}[\/latex] the series is geometric with a common ratio of [latex]\\frac{1}{3}\\text{.}[\/latex] The sum of the infinite series is defined.<\/li>\n<li>The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-574\">Determine whether the sum of the infinite series is defined.<\/p>\n<div id=\"fs-id1165135497153\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137619661\">\n<div id=\"fs-id1165137619663\">\n<p id=\"fs-id1165137619666\">[latex]\\frac{1}{3}+\\frac{1}{2}+\\frac{3}{4}+\\frac{9}{8}+...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137823177\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137823179\">The sum is not defined.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-827\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_10\">\n<div id=\"fs-id1165137627905\">\n<p id=\"fs-id1165137627907\">[latex]24+\\left(-12\\right)+6+\\left(-3\\right)+...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137400663\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137400665\">The sum of the infinite series is defined.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-662\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_11\">\n<div id=\"fs-id1165137400674\">\n<p id=\"fs-id1165135159927\">[latex]\\sum _{k=1}^{\\infty }15\\cdot {\\left(\u20130.3\\right)}^{k}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135437872\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135437874\">The sum of the infinite series is defined.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437882\" class=\"bc-section section\">\n<h4>Finding Sums of Infinite Series<\/h4>\n<p id=\"fs-id1165137679221\">When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first [latex]n[\/latex]terms of a geometric series.<\/p>\n<div id=\"fs-id1165135208470\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex]<\/div>\n<p id=\"fs-id1165137552673\">We will examine an infinite series with [latex]r=\\frac{1}{2}.[\/latex] What happens to [latex]{r}^{n}[\/latex] as [latex]n[\/latex] increases?<\/p>\n<div id=\"fs-id1165131961708\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(\\frac{1}{2}\\right)}^{2}=\\frac{1}{4}\\\\ {\\left(\\frac{1}{2}\\right)}^{3}=\\frac{1}{8}\\\\ {\\left(\\frac{1}{2}\\right)}^{4}=\\frac{1}{16}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137725370\">The value of[latex]\\,{r}^{n}\\,[\/latex]decreases rapidly. What happens for greater values of [latex]n?[\/latex]<\/p>\n<div id=\"fs-id1165135655357\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(\\frac{1}{2}\\right)}^{10}=\\frac{1}{1\\text{,}024}\\hfill \\\\ {\\left(\\frac{1}{2}\\right)}^{20}=\\frac{1}{1\\text{,}048\\text{,}576}\\hfill \\\\ {\\left(\\frac{1}{2}\\right)}^{30}=\\frac{1}{1\\text{,}073\\text{,}741\\text{,}824}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137673611\">As [latex]n[\/latex] gets very large, [latex]{r}^{n}[\/latex] gets very small. We say that, as [latex]n[\/latex] increases without bound, [latex]{r}^{n}[\/latex]approaches 0. As [latex]{r}^{n}[\/latex] approaches 0,[latex]1-{r}^{n}[\/latex] approaches 1. When this happens, the numerator approaches[latex]\\,{a}_{1}.[\/latex] This give us a formula for the sum of an infinite geometric series.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Formula for the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165135187536\">The formula for the sum of an infinite geometric series with [latex]-1<r<1[\/latex] is<\/p>\n<div id=\"fs-id1165137851356\">[latex]S=\\frac{{a}_{1}}{1-r}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135190124\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137431516\"><strong>Given an infinite geometric series, find its sum.<\/strong><\/p>\n<ol id=\"fs-id1165137431522\" type=\"1\">\n<li>Identify[latex]{a}_{1}[\/latex]and [latex]r.[\/latex]<\/li>\n<li>Confirm that [latex]\u20131<r<1.[\/latex]<\/li>\n<li>Substitute values for [latex]{a}_{1}[\/latex] and [latex]r[\/latex] into the formula, [latex]S=\\frac{{a}_{1}}{1-r}.[\/latex]<\/li>\n<li>Simplify to find[latex]\\,S.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137673641\">\n<div id=\"fs-id1165137673643\">\n<h3>Finding the Sum of an Infinite Geometric Series<\/h3>\n<p id=\"fs-id1165137832084\">Find the sum, if it exists, for the following:<\/p>\n<ol id=\"fs-id1165137832087\" type=\"a\">\n<li>[latex]10+9+8+7+\\dots[\/latex]<\/li>\n<li>[latex]248.6+99.44+39.776+\\text{ }\\dots[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{\\infty }4\\text{,}374\\cdot {\\left(\u2013\\frac{1}{3}\\right)}^{k\u20131}[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{\\infty }\\frac{1}{9}\\cdot {\\left(\\frac{4}{3}\\right)}^{k}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135394090\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135394092\" type=\"a\">\n<li>There is not a constant ratio; the series is not geometric.<\/li>\n<li>\n<p id=\"fs-id1165135252107\">There is a constant ratio; the series is geometric. [latex]{a}_{1}=248.6[\/latex]and[latex]r=\\frac{99.44}{248.6}=0.4,[\/latex] so the sum exists. Substitute [latex]{a}_{1}=248.6[\/latex] and [latex]r=0.4[\/latex] into the formula and simplify to find the sum:<\/p>\n<div id=\"fs-id1165134220772\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}S=\\frac{{a}_{1}}{1-r}\\hfill \\\\ S=\\frac{248.6}{1-0.4}=414.\\overline{3}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165135242798\">The formula is exponential, so the series is geometric with [latex]r=\u2013\\frac{1}{3}.[\/latex] Find[latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula:<\/p>\n<div id=\"fs-id1165134284441\" class=\"unnumbered aligncenter\">[latex]{a}_{1}=4\\text{,}374\\cdot {\\left(\u2013\\frac{1}{3}\\right)}^{1\u20131}=4\\text{,}374[\/latex]<\/div>\n<p id=\"fs-id1165135203509\">Substitute [latex]{a}_{1}=4\\text{,}374[\/latex] and [latex]r=-\\frac{1}{3}[\/latex] into the formula, and simplify to find the sum:<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}S=\\frac{{a}_{1}}{1-r}\\hfill \\\\ S=\\frac{4\\text{,}374}{1-\\left(-\\frac{1}{3}\\right)}=3\\text{,}280.5\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>The formula is exponential, so the series is geometric, but[latex]\\,r>1.\\,[\/latex]The sum does not exist.<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_04_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137737059\">\n<div id=\"fs-id1165137737061\">\n<h3>Finding an Equivalent Fraction for a Repeating Decimal<\/h3>\n<p id=\"fs-id1165135160513\">Find an equivalent fraction for the repeating decimal [latex]0.\\overline{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135512477\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135512479\">We notice the repeating decimal [latex]0.\\overline{3}=0.333...[\/latex] so we can rewrite the repeating decimal as a sum of terms.<\/p>\n<div id=\"fs-id1165135177536\" class=\"unnumbered aligncenter\">[latex]0.\\overline{3}=0.3+0.03+0.003+...[\/latex]<\/div>\n<p id=\"fs-id1165137761250\">Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.<\/p>\n<p><span id=\"eip-id1164269283321\"><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154917\/CNX_Precalc_Figure_11_04_002.jpg\" alt=\"...\" \/><\/span><\/p>\n<p id=\"fs-id1165135195706\">Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have<\/p>\n<div id=\"fs-id1165135575202\" class=\"unnumbered aligncenter\">[latex]{S}_{n}=\\frac{{a}_{1}}{1-r}=\\frac{0.3}{1-0.1}=\\frac{0.3}{0.9}=\\frac{1}{3}.[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-120\">Find the sum, if it exists.<\/p>\n<div id=\"fs-id1165135208424\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_12\">\n<div id=\"fs-id1165137446138\">\n<p id=\"fs-id1165137446140\">[latex]2+\\frac{2}{3}+\\frac{2}{9}+...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137834920\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137834922\">3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1387862\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_13\">\n<div id=\"fs-id1165137834931\">\n<p id=\"fs-id1165137834933\">[latex]\\sum _{k=1}^{\\infty }0.76k+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135596445\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135596447\">The series is not geometric.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1387940\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_14\">\n<div id=\"fs-id1165134042630\">\n<p id=\"fs-id1165134042632\">[latex]\\sum _{k=1}^{\\infty }{\\left(-\\frac{3}{8}\\right)}^{k}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137387166\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137387168\">[latex]-\\frac{3}{11}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390954\" class=\"bc-section section\">\n<h3>Solving Annuity Problems<\/h3>\n<p id=\"fs-id1165137794312\">At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% <span class=\"no-emphasis\">annual interest<\/span>, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.<\/p>\n<p id=\"fs-id1165134186203\">We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5%=1.005.[\/latex] After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.<\/p>\n<p id=\"fs-id1165135191421\">We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72.[\/latex] We can substitute [latex]{a}_{1}=50, r=1.005, \\text{and} n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.<\/p>\n<div id=\"fs-id1165135531315\" class=\"unnumbered aligncenter\">[latex]{S}_{72}=\\frac{50\\left(1-{1.005}^{72}\\right)}{1-1.005}\\approx 4\\text{,}320.44[\/latex]<\/div>\n<p id=\"fs-id1165135510768\">After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of <strong>72(50) = $3,600<\/strong>.&nbsp; This means that because of the annuity, the couple earned $720.44 interest in their college fund.<\/p>\n<div id=\"fs-id1165135510774\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135342192\"><strong>Given an initial deposit and an interest rate, find the value of an annuity.<\/strong><\/p>\n<ol type=\"1\">\n<li>Determine[latex]\\,{a}_{1}\\text{,}\\,[\/latex]the value of the initial deposit.<\/li>\n<li>Determine[latex]\\,n\\text{,}\\,[\/latex]the number of deposits.<\/li>\n<li>Determine[latex]\\,r.[\/latex]\n<ol id=\"fs-id1165135187090\" type=\"a\">\n<li>Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\n<li>Add 1 to this amount to find [latex]r.[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Substitute values for[latex]\\,{a}_{1}\\text{,}\\,r,\\,\\text{and}\\,n\\,[\/latex]<br \/>\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series,[latex]{S}_{n}=\\frac{{a}_{1}\\left(1\u2013{r}^{n}\\right)}{1\u2013r}.[\/latex]<\/li>\n<li>Simplify to find [latex]{S}_{n},[\/latex] the value of the annuity after [latex]n[\/latex] deposits.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_04_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135173286\">\n<div id=\"fs-id1165135173288\">\n<h3>Solving an Annuity Problem<\/h3>\n<p>A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?<\/p>\n<\/div>\n<div id=\"fs-id1165135388500\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135388502\">The value of the initial deposit is $100, so[latex]\\,{a}_{1}=100.\\,[\/latex]A total of 120 monthly deposits are made in the 10 years, so[latex]n=120.[\/latex] To find [latex]r,\\,[\/latex]divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.<\/p>\n<div class=\"unnumbered\">[latex]r=1+\\frac{0.09}{12}=1.0075[\/latex]<\/div>\n<p id=\"fs-id1165137722693\">Substitute[latex]\\,{a}_{1}=100\\text{,}\\,r=1.0075\\text{,}\\,\\text{and}\\,n=120\\,[\/latex]into the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, and simplify to find the value of the annuity.<\/p>\n<div id=\"fs-id1165135582181\" class=\"unnumbered aligncenter\">[latex]{S}_{120}=\\frac{100\\left(1-{1.0075}^{120}\\right)}{1-1.0075}\\approx 19\\text{,}351.43[\/latex]<\/div>\n<p id=\"fs-id1165135503815\">So the account has $19,351.43 after the last deposit is made.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503821\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_04_15\">\n<div id=\"fs-id1165135503830\">\n<p id=\"fs-id1165137725104\">At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?<\/p>\n<\/div>\n<div id=\"fs-id1165137725110\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137725113\">$32,775.87<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137725119\" class=\"precalculus media\">\n<p id=\"fs-id1165135188268\">Access these online resources for additional instruction and practice with series.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/arithmeticser\">Arithmetic Series<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/geometricser\">Geometric Series<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/sumnotation\">Summation Notation<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137740973\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134342462\" summary=\"..\">\n<tbody>\n<tr>\n<td>sum of the first[latex]\\,n\\,[\/latex]<br \/>\nterms of an arithmetic series<\/td>\n<td>[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of the first[latex]\\,n\\,[\/latex]<br \/>\nterms of a geometric series<\/td>\n<td>[latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r},r\\ne 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of an infinite geometric series with[latex]\\,\u20131<r<\\text{ }1[\/latex]<\/td>\n<td>[latex]{S}_{n}=\\frac{{a}_{1}}{1-r},r\\ne 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137834434\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137834440\">\n<li>The sum of the terms in a sequence is called a series.<\/li>\n<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See <a class=\"autogenerated-content\" href=\"#Example_11_04_01\">(Figure)<\/a>.<\/li>\n<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\n<li>The sum of the first[latex]n[\/latex]terms of an arithmetic series can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_04_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_04_03\">(Figure)<\/a>.<\/li>\n<li>The sum of the terms in a geometric sequence is called a geometric series.<\/li>\n<li>The sum of the first[latex]n[\/latex]terms of a geometric series can be found using a formula. See <a class=\"autogenerated-content\" href=\"#Example_11_04_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_04_05\">(Figure)<\/a>.<\/li>\n<li>The sum of an infinite series exists if the series is geometric with [latex]\u20131<r<1.[\/latex]<\/li>\n<li>If the sum of an infinite series exists, it can be found using a formula. See <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_06\">(Figure)<\/a>, <\/strong><a class=\"autogenerated-content\" href=\"#Example_11_04_07\">(Figure)<\/a><strong>, <\/strong>and <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_08\">(Figure)<\/a><\/strong>.<\/li>\n<li>An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See <strong><a class=\"autogenerated-content\" href=\"#Example_11_04_09\">(Figure)<\/a><\/strong>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134109732\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135639904\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135639910\">\n<div id=\"fs-id1165135639912\">\n<p id=\"fs-id1165135639914\">What is an [latex]n\\text{th}[\/latex] partial sum?<\/p>\n<\/div>\n<div id=\"fs-id1165134387624\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134387626\">An [latex]n\\text{th}[\/latex] partial sum is the sum of the first [latex]n[\/latex] terms of a sequence.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635137\">\n<div id=\"fs-id1165137635139\">\n<p id=\"fs-id1165135435776\">What is the difference between an arithmetic sequence and an arithmetic series?<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135435783\">\n<p id=\"fs-id1165135435785\">What is a geometric series?<\/p>\n<\/div>\n<div id=\"fs-id1165135435790\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135435792\">A geometric series is the sum of the terms in a geometric sequence.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693460\">\n<div id=\"fs-id1165137693462\">\n<p id=\"fs-id1165137693464\">How is finding the sum of an infinite geometric series different from finding the [latex]n\\text{th}[\/latex] partial sum?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487089\">\n<div id=\"fs-id1165135487091\">\n<p id=\"fs-id1165135487093\">What is an annuity?<\/p>\n<\/div>\n<div id=\"fs-id1165135487097\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135193165\">An annuity is a series of regular equal payments that earn a constant compounded interest.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193172\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135193177\">For the following exercises, express each description of a sum using summation notation.<\/p>\n<div id=\"fs-id1165135193180\">\n<div id=\"fs-id1165135193182\">\n<p id=\"fs-id1165135187266\">The sum of terms [latex]{m}^{2}+3m[\/latex]from [latex]m=1[\/latex] to [latex]m=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137705183\">\n<p id=\"fs-id1165137705186\">The sum from of [latex]n=0[\/latex] to [latex]n=4[\/latex] of [latex]5n[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135205629\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135205630\">[latex]\\sum _{n=0}^{4}5n[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137464457\">\n<div id=\"fs-id1165137464459\">\n<p id=\"fs-id1165137464461\">The sum of [latex]6k-5[\/latex] from [latex]k=-2[\/latex] to [latex]k=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333032\">\n<div id=\"fs-id1165135333034\">\n<p id=\"fs-id1165135333036\">The sum that results from adding the number 4 five times<\/p>\n<\/div>\n<div id=\"fs-id1165135333040\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135333042\">[latex]\\sum _{k=1}^{5}4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135168238\">For the following exercises, express each arithmetic sum using summation notation.<\/p>\n<div id=\"fs-id1165134149128\">\n<div id=\"fs-id1165134149130\">\n<p id=\"fs-id1165134149132\">[latex]5+10+15+20+25+30+35+40+45+50[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135404219\">\n<div id=\"fs-id1165135190494\">\n<p id=\"fs-id1165135190496\">[latex]10+18+26+\\dots +162[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137696462\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\sum _{k=1}^{20}8k+2[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194435\">\n<div id=\"fs-id1165135194437\">\n<p id=\"fs-id1165135194440\">[latex]\\frac{1}{2}+1+\\frac{3}{2}+2+\\dots +4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137770272\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each arithmetic sequence.<\/p>\n<div>\n<div id=\"fs-id1165137770283\">[latex]\\frac{3}{2}+2+\\frac{5}{2}+3+\\frac{7}{2}[\/latex]<\/div>\n<div id=\"fs-id1165137871515\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137871517\">[latex]{S}_{5}=\\frac{5\\left(\\frac{3}{2}+\\frac{7}{2}\\right)}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613235\">\n<div id=\"fs-id1165135613237\">\n<p id=\"fs-id1165135613239\">[latex]19+25+31+\\dots +73[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137745115\">\n<div id=\"fs-id1165137745117\">\n<p id=\"fs-id1165137745119\">[latex]3.2+3.4+3.6+\\dots +5.6[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137619905\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137834864\">[latex]{S}_{13}=\\frac{13\\left(3.2+5.6\\right)}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135569954\">For the following exercises, express each geometric sum using summation notation.<\/p>\n<div id=\"fs-id1165135705038\">\n<div id=\"fs-id1165135705040\">\n<p id=\"fs-id1165135705042\">[latex]1+3+9+27+81+243+729+2187[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137817670\">\n<div id=\"fs-id1165137817672\">[latex]8+4+2+\\dots +0.125[\/latex]<\/div>\n<div id=\"fs-id1165137456659\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137456662\">[latex]\\sum _{k=1}^{7}8\\cdot {0.5}^{k-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137769967\">\n<div id=\"fs-id1165137769969\">\n<p id=\"fs-id1165137769971\">[latex]-\\frac{1}{6}+\\frac{1}{12}-\\frac{1}{24}+\\dots +\\frac{1}{768}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135203486\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each geometric sequence, and then state the indicated sum.<\/p>\n<div id=\"fs-id1165135203490\">\n<div id=\"fs-id1165135203492\">\n<p id=\"fs-id1165135203495\">[latex]9+3+1+\\frac{1}{3}+\\frac{1}{9}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135338212\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135338214\">[latex]{S}_{5}=\\frac{9\\left(1-{\\left(\\frac{1}{3}\\right)}^{5}\\right)}{1-\\frac{1}{3}}=\\frac{121}{9}\\approx 13.44[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137565170\">\n<div id=\"fs-id1165137565172\">\n<p id=\"fs-id1165137565174\">[latex]\\sum _{n=1}^{9}5\\cdot {2}^{n-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503927\">\n<div id=\"fs-id1165135503929\">\n<p id=\"fs-id1165135503931\">[latex]\\sum _{a=1}^{11}64\\cdot {0.2}^{a-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137704468\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135205888\">[latex]{S}_{11}=\\frac{64\\left(1-{0.2}^{11}\\right)}{1-0.2}=\\frac{781,249,984}{9,765,625}\\approx 80[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135417823\">For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.<\/p>\n<div id=\"fs-id1165135417827\">\n<div id=\"fs-id1165137741039\">\n<p id=\"fs-id1165137741041\">[latex]12+18+24+30+...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137862580\">[latex]2+1.6+1.28+1.024+...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135496306\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135496308\">The series is defined. [latex]S=\\frac{2}{1-0.8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333611\">\n<div id=\"fs-id1165135333613\">\n<p id=\"fs-id1165135333615\">[latex]\\sum _{m=1}^{\\infty }{4}^{m-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571886\">\n<div id=\"fs-id1165135571888\">[latex]\\underset{\\infty }{\\overset{k=1}{{\\sum }^{\\text{\u200b}}}}-{\\left(-\\frac{1}{2}\\right)}^{k-1}[\/latex]<\/div>\n<div id=\"fs-id1165137701119\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135408472\">The series is defined. [latex]S=\\frac{-1}{1-\\left(-\\frac{1}{2}\\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827903\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137894472\">For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.<\/p>\n<div id=\"fs-id1165137894476\">\n<div id=\"fs-id1165137894478\">\n<p>Graph the arithmetic sequence showing one year of Javier\u2019s deposits.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137894486\">\n<div id=\"fs-id1165137894488\">\n<p id=\"fs-id1165137828073\">Graph the arithmetic series showing the monthly sums of one year of Javier\u2019s deposits.<\/p>\n<\/div>\n<div id=\"fs-id1165137828079\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137828084\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154923\/CNX_Precalc_Figure_11_04_202.jpg\" alt=\"Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137827820\">For the following exercises, use the geometric series[latex]{\\sum _{k=1}^{\\infty }\\left(\\frac{1}{2}\\right)}^{k}.[\/latex]<\/p>\n<div id=\"fs-id1165137761311\">\n<div id=\"fs-id1165137761314\">\n<p id=\"fs-id1165137761316\">Graph the first 7 partial sums of the series.<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137642999\">\n<p id=\"fs-id1165137643001\">What number does [latex]{S}_{n}[\/latex] seem to be approaching in the graph? Find the sum to explain why this makes sense.<\/p>\n<\/div>\n<div id=\"fs-id1165135208860\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135208862\">Sample answer: The graph of [latex]{S}_{n}[\/latex] seems to be approaching 1. This makes sense because[latex]\\sum _{k=1}^{\\infty }{\\left(\\frac{1}{2}\\right)}^{k}[\/latex]is a defined infinite geometric series with [latex]S=\\frac{\\frac{1}{2}}{1\u2013\\left(\\frac{1}{2}\\right)}=1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203672\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135701517\">For the following exercises, find the indicated sum.<\/p>\n<div id=\"fs-id1165135701521\">\n<div id=\"fs-id1165135701523\">\n<p id=\"fs-id1165135701525\">[latex]\\sum _{a=1}^{14}a[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891293\">\n<div id=\"fs-id1165137891295\">\n<p id=\"fs-id1165137891297\">[latex]\\sum _{n=1}^{6}n\\left(n-2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137453619\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137453621\">49<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453626\">\n<div id=\"fs-id1165137453629\">[latex]\\sum _{k=1}^{17}{k}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137736016\">\n<div id=\"fs-id1165137736018\">\n<p id=\"fs-id1165137736020\">[latex]\\sum _{k=1}^{7}{2}^{k}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134183115\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134183117\">254<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134183122\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series to find the sum.<\/p>\n<div id=\"fs-id1165135194113\">\n<div id=\"fs-id1165135194116\">\n<p id=\"fs-id1165135194118\">[latex]-1.7+-0.4+0.9+2.2+3.5+4.8[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723368\">\n<div id=\"fs-id1165137807476\">\n<p id=\"fs-id1165137807478\">[latex]6+\\frac{15}{2}+9+\\frac{21}{2}+12+\\frac{27}{2}+15[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135471078\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135471080\">[latex]{S}_{7}=\\frac{147}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042711\">\n<div id=\"fs-id1165134042713\">\n<p id=\"fs-id1165137673868\">[latex]-1+3+7+...+31[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849073\">\n<div id=\"fs-id1165137849075\">[latex]\\sum _{k=1}^{11}\\left(\\frac{k}{2}-\\frac{1}{2}\\right)[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{S}_{11}=\\frac{55}{2}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135301093\">For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of a geometric series to find the partial sum.<\/p>\n<div id=\"fs-id1165135516941\">\n<div id=\"fs-id1165135516943\">\n<p>[latex]{S}_{6}[\/latex] for the series [latex]-2-10-50-250...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137784646\">[latex]{S}_{7}[\/latex] for the series [latex]0.4-2+10-50...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137530296\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137530298\">[latex]{S}_{7}=5208.4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165135187164\">[latex]\\sum _{k=1}^{9}{2}^{k-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134094671\">\n<div id=\"fs-id1165134094673\">\n<p id=\"fs-id1165134094676\">[latex]\\sum _{n=1}^{10}-2\\cdot {\\left(\\frac{1}{2}\\right)}^{n-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135496397\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135496400\">[latex]{S}_{10}=-\\frac{1023}{256}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137749444\">For the following exercises, find the sum of the infinite geometric series.<\/p>\n<div id=\"fs-id1165137749447\">\n<div id=\"fs-id1165137749449\">\n<p id=\"fs-id1165137749451\">[latex]4+2+1+\\frac{1}{2}...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137738012\">\n<div id=\"fs-id1165137738014\">\n<p id=\"fs-id1165137761900\">[latex]-1-\\frac{1}{4}-\\frac{1}{16}-\\frac{1}{64}...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137535603\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137535605\">[latex]S=-\\frac{4}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135332680\">\n<div id=\"fs-id1165135332682\">\n<p id=\"fs-id1165135332684\">[latex]\\underset{\\infty }{\\overset{k=1}{{\\sum }^{\\text{\u200b}}}}3\\cdot {\\left(\\frac{1}{4}\\right)}^{k-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734577\">\n<div id=\"fs-id1165137734579\">\n<p id=\"fs-id1165135342830\">[latex]\\sum _{n=1}^{\\infty }4.6\\cdot {0.5}^{n-1}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135333829\">[latex]S=9.2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137556920\">For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.<\/p>\n<div id=\"fs-id1165137556925\">\n<div>\n<p id=\"fs-id1165137556929\">Deposit amount: [latex]\\text{\\$}50;[\/latex] total deposits: [latex]60;[\/latex] interest rate: [latex]5%,[\/latex] compounded monthly<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205791\">\n<div id=\"fs-id1165135205793\">\n<p id=\"fs-id1165135205796\">Deposit amount: [latex]\\text{\\$}150;[\/latex] total deposits: [latex]24;[\/latex] interest rate: [latex]3%,[\/latex] compounded monthly<\/p>\n<\/div>\n<div id=\"fs-id1165137705498\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137705500\">$3,705.42<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705506\">\n<div id=\"fs-id1165137705508\">\n<p id=\"fs-id1165137705510\">Deposit amount: [latex]\\text{\\$}450;[\/latex] total deposits: [latex]60;[\/latex] interest rate: [latex]4.5%,[\/latex] compounded quarterly<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134157171\">\n<div id=\"fs-id1165134157173\">\n<p id=\"fs-id1165134157175\">Deposit amount: [latex]\\text{\\$}100;[\/latex] total deposits: [latex]120;[\/latex] interest rate: [latex]10%,[\/latex] compounded semi-annually<\/p>\n<\/div>\n<div id=\"fs-id1165135481987\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135481990\">$695,823.97<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135481996\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135482001\">\n<div id=\"fs-id1165135422912\">\n<p id=\"fs-id1165135422915\">The sum of terms [latex]50-{k}^{2}[\/latex] from [latex]k=x[\/latex] through [latex]7[\/latex] is [latex]115.[\/latex] What is <em>x<\/em>?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188088\">\n<div id=\"fs-id1165135188090\">\n<p id=\"fs-id1165135188092\">Write an explicit formula for[latex]{a}_{k}[\/latex]such that[latex]\\sum _{k=0}^{6}{a}_{k}=189.[\/latex] Assume this is an arithmetic series.<\/p>\n<\/div>\n<div id=\"fs-id1165135245742\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135245744\">[latex]{a}_{k}=30-k[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134212315\">\n<div id=\"fs-id1165134212317\">\n<p id=\"fs-id1165134212319\">Find the smallest value of <em>n<\/em> such that[latex]\\sum _{k=1}^{n}\\left(3k\u20135\\right)>100.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135500068\">\n<div id=\"fs-id1165135500070\">\n<p id=\"fs-id1165135500072\">How many terms must be added before the series [latex]-1-3-5-7....\\text{ }[\/latex]has a sum less than [latex]-75?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134061152\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134061154\">9 terms<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134061159\">\n<div id=\"fs-id1165134061161\">\n<p id=\"fs-id1165134061163\">Write [latex]0.\\overline{65}[\/latex] as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert [latex]0.\\overline{65}[\/latex] to a fraction.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134406874\">\n<div id=\"fs-id1165134406876\">\n<p id=\"fs-id1165137817620\">The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?<\/p>\n<\/div>\n<div id=\"fs-id1165137817626\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137817628\">[latex]r=\\frac{4}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191488\">\n<div id=\"fs-id1165135191490\">\n<p id=\"fs-id1165135191492\">To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191500\">\n<div id=\"fs-id1165135191503\">\n<p id=\"fs-id1165135191505\">Karl has two years to save [latex]$10,000[\/latex] to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?<\/p>\n<\/div>\n<div id=\"fs-id1165135181468\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135181470\">$400 per month<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187471\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135187477\">\n<div id=\"fs-id1165135187479\">\n<p id=\"fs-id1165135187481\">Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for[latex]\\,1\\,[\/latex]hour, and each successive day she will increase her study time by[latex]\\,30\\,[\/latex]minutes. How many hours will Keisha have studied after one week?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181425\">\n<div id=\"fs-id1165135181427\">\n<p id=\"fs-id1165135181429\">A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?<\/p>\n<\/div>\n<div id=\"fs-id1165135181435\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135181437\">420 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181442\">\n<div id=\"fs-id1165135181445\">\n<p id=\"fs-id1165137696121\">A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696127\">\n<div id=\"fs-id1165137696130\">\n<p id=\"fs-id1165137696132\">A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels [latex]\\frac{3}{4}[\/latex] the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?<\/p>\n<\/div>\n<div id=\"fs-id1165137629057\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137629059\">12 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137629064\">\n<div id=\"fs-id1165137629067\">\n<p id=\"fs-id1165137629069\">Rachael deposits $1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137726792\">\n<dt>annuity<\/dt>\n<dd id=\"fs-id1165137726797\">an investment in which the purchaser makes a sequence of periodic, equal payments<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137726801\">\n<dt>arithmetic series<\/dt>\n<dd id=\"fs-id1165137726806\">the sum of the terms in an arithmetic sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137726811\">\n<dt>diverge<\/dt>\n<dd>a series is said to diverge if the sum is not a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134031386\">\n<dt>geometric series<\/dt>\n<dd id=\"fs-id1165134031391\">the sum of the terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134031396\">\n<dt>index of summation<\/dt>\n<dd id=\"fs-id1165134031401\">in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation<\/dd>\n<\/dl>\n<dl>\n<dt>infinite series<\/dt>\n<dd id=\"fs-id1165137737867\">the sum of the terms in an infinite sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137737872\">\n<dt>lower limit of summation<\/dt>\n<dd id=\"fs-id1165137737877\">the number used in the explicit formula to find the first term in a series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137737881\">\n<dt>nth partial sum<\/dt>\n<dd id=\"fs-id1165135471113\">the sum of the first[latex]n[\/latex]terms of a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135471124\">\n<dt>series<\/dt>\n<dd id=\"fs-id1165135471129\">the sum of the terms in a sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135471133\">\n<dt>summation notation<\/dt>\n<dd id=\"fs-id1165135471138\">a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762744\">\n<dt>upper limit of summation<\/dt>\n<dd id=\"fs-id1165137762749\">the number used in the explicit formula to find the last term in a series<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":5,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-207","chapter","type-chapter","status-publish","hentry"],"part":198,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/207\/revisions"}],"predecessor-version":[{"id":208,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/207\/revisions\/208"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/198"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/207\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=207"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=207"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=207"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}