{"id":1861,"date":"2023-10-12T00:32:21","date_gmt":"2023-10-12T00:32:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-geometric-sequences\/"},"modified":"2023-10-12T00:32:21","modified_gmt":"2023-10-12T00:32:21","slug":"introduction-geometric-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-geometric-sequences\/","title":{"raw":"Geometric Sequences &amp; Series","rendered":"Geometric Sequences &amp; Series"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n \t<li class=\"li2\"><span class=\"s1\">Find the common ratio for a geometric sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">List the terms of a geometric sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use a recursive formula for a geometric sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use an explicit formula for a geometric sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of a geometric series.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of an in\ufb01nite geometric series.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Solve annuity problems.<\/span><\/li>\n<\/ul>\n<\/div>\nMany jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section we will review sequences that grow in this way.\n<h2>Terms of Geometric Sequences<\/h2>\n<h3>Finding Common Ratios<\/h3>\nThe yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223627\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\">\n<div class=\"textbox\">\n<h3>A General Note: Definition of a Geometric Sequence<\/h3>\nA <strong>geometric sequence<\/strong> is one in which any term divided by the previous term is a constant. This constant is called the <strong>common ratio<\/strong> of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/h3>\n<ol>\n \t<li>Divide each term by the previous term.<\/li>\n \t<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Common Ratios<\/h3>\nIs the sequence geometric? If so, find the common ratio.\n<ol>\n \t<li>[latex]1,2,4,8,16,\\dots[\/latex]<\/li>\n \t<li>[latex]48,12,4,2,\\dots[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"706624\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"706624\"]\n\nDivide each term by the previous term to determine whether a common ratio exists.\n<ol>\n \t<li>[latex]\\begin{align}&amp;\\frac{2}{1}=2 &amp;&amp; \\frac{4}{2}=2 &amp;&amp; \\frac{8}{4}=2 &amp;&amp; \\frac{16}{8}=2 \\end{align}[\/latex]\nThe sequence is geometric because there is a common ratio. The common ratio is 2.<\/li>\n \t<li>[latex]\\begin{align}&amp;\\frac{12}{48}=\\frac{1}{4} &amp;&amp; \\frac{4}{12}=\\frac{1}{3} &amp;&amp; \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]\nThe sequence is not geometric because there is not a common ratio.<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\nThe graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223630\/CNX_Precalc_Figure_11_03_0022.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\">[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/h4>\n<em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nIs the sequence geometric? If so, find the common ratio.\n[latex]5,10,15,20,\\dots[\/latex]\n\n[reveal-answer q=\"893960\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"893960\"]\n\nThe sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .\n\n[\/hidden-answer]\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68722&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nIs the sequence geometric? If so, find the common ratio.\n\n[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]\n\n[reveal-answer q=\"870692\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"870692\"]\n\nThe sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .\n\n[\/hidden-answer]\n\n<\/div>\n<h3>Writing Terms of Geometric Sequences<\/h3>\nNow that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{1}=-2 \\\\ &amp;{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &amp;{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &amp;{a}_{4}=\\left(-32\\cdot 4\\right)=-128 \\end{align}[\/latex]<\/p>\nThe first four terms are [latex]\\left\\{-2,-8,-32,-128\\right\\}[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n<ol>\n \t<li>Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n \t<li>Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n \t<li>Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Terms of a Geometric Sequence<\/h3>\nList the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].\n\n[reveal-answer q=\"650557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"650557\"]\n\nMultiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],&nbsp;and so on.\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{1}=5 \\\\ &amp;{a}_{2}=-2{a}_{1}=-10 \\\\ &amp;{a}_{3}=-2{a}_{2}=20 \\\\ &amp;{a}_{4}=-2{a}_{3}=-40 \\end{align}[\/latex]<\/p>\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\dfrac{1}{3}[\/latex].\n\n[reveal-answer q=\"228021\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"228021\"]\n\n[latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156689&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<h2>Explicit Formulas for Geometric Sequences<\/h2>\n<h3>Using Explicit Formulas for Geometric Sequences<\/h3>\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/p>\nLet\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is\n<p style=\"text-align: center;\">[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/p>\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223634\/CNX_Precalc_Figure_11_03_0042.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\">\n<div class=\"textbox\">\n<h3>A General Note: Explicit Formula for a Geometric Sequence<\/h3>\nThe <em>n<\/em>th term of a geometric sequence is given by the <strong>explicit formula<\/strong>:\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\nGiven a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].\n\n[reveal-answer q=\"602906\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"602906\"]\n\nThe sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].\n<p style=\"text-align: center;\">[latex]3,3r,3{r}^{2},3{r}^{3},\\dots[\/latex]<\/p>\nFind the common ratio using the given fourth term.\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}{r}^{n - 1} \\\\ &amp;{a}_{4}=3{r}^{3} &amp;&amp; \\text{Write the fourth term of sequence in terms of }{a}_{1}\\text{ and }r \\\\ &amp;24=3{r}^{3} &amp;&amp; \\text{Substitute }24\\text{ for }{a}_{4} \\\\ &amp;8={r}^{3} &amp;&amp; \\text{Divide} \\\\ &amp;r=2 &amp;&amp; \\text{Solve for the common ratio} \\end{align}[\/latex]<\/p>\nFind the second term by multiplying the first term by the common ratio.\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{2} &amp; =2{a}_{1} \\\\ &amp; =2\\left(3\\right) \\\\ &amp; =6 \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nGiven a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].\n\n[reveal-answer q=\"132711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"132711\"]\n\n[latex]{a}_{6}=16\\text{,}384[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5856&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\nWrite an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.\n<p style=\"text-align: center;\">[latex]\\left\\{2,10,50,250,\\dots\\right\\}[\/latex]<\/p>\n[reveal-answer q=\"202402\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"202402\"]\n\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\n<p style=\"text-align: center;\">[latex]\\dfrac{10}{2}=5[\/latex]<\/p>\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)} \\\\ &amp;{a}_{n}=2\\cdot {5}^{n - 1} \\end{align}[\/latex]<\/p>\nThe graph of this sequence shows an exponential pattern.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223636\/CNX_Precalc_Figure_11_03_0052.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite an explicit formula for the following geometric sequence.\n<p style=\"text-align: center;\">[latex]\\left\\{-1,3,-9,27,\\dots\\right\\}[\/latex]<\/p>\n[reveal-answer q=\"521311\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"521311\"]\n\n[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29756&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\nIn real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we can alter the explicit formula slightly by using the following formula:\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/p>\n\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Geometric Sequences<\/h3>\nIn 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.\n<ol>\n \t<li>Write a formula for the student population.<\/li>\n \t<li>Estimate the student population in 2020.<\/li>\n<\/ol>\n[reveal-answer q=\"304368\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"304368\"]\n<ol>\n \t<li>The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get\n[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/li>\n \t<li>We can find the number of years since 2013 by subtracting.\n[latex]2020 - 2013=7[\/latex]\nWe are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.\n[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]\nThe student population will be about 374 in 2020.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.\n<ol style=\"list-style-type: lower-alpha;\">\n \t<li>Write a formula for the number of hits.<\/li>\n \t<li>Estimate the number of hits in 5 weeks.<\/li>\n<\/ol>\n[reveal-answer q=\"67595\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"67595\"]\n<ol style=\"list-style-type: lower-alpha;\">\n \t<li>[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]<\/li>\n \t<li>The number of hits will be about 333.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\nThe following video provides a short lesson on some of the topics covered in this lesson.\n\nhttps:\/\/youtu.be\/XHyeLKZYb2w\n\n&nbsp;\n\nA 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.\n<h2>Geometric Series<\/h2>\nJust 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 <strong>geometric sequence<\/strong> is a sequence in which the ratio of any two consecutive terms is the <strong>common ratio<\/strong>, [latex]r[\/latex]. We can write the sum of the first [latex]n[\/latex] terms of a geometric series as\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1}[\/latex].<\/p>\nJust 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].\n<p style=\"text-align: center;\">[latex]r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}[\/latex]<\/p>\nNext, we subtract this equation from the original equation.\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&amp;={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1} \\\\ -r{S}_{n}&amp;=-\\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}\\right) \\\\ \\hline \\left(1-r\\right){S}_{n}&amp;={a}_{1}-{a}_{1}{r}^{n}\\end{align}[\/latex]<\/p>\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], factor [latex]a_1[\/latex] on the right hand side and divide both sides by [latex]\\left(1-r\\right)[\/latex].\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Sum of the First <em>n<\/em> Terms of a Geometric Series<\/h3>\nA <strong>geometric series<\/strong> 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\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a geometric series, find the sum of the first <em>n<\/em> terms.<\/h3>\n<ol>\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}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\n \t<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: 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>\n \t<li>[latex]{S}_{11}[\/latex] for the series [latex] 8 + -4 + 2 + \\dots [\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^6 3\\cdot {2}^{k}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"618333\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"618333\"]\n<ol>\n \t<li>[latex]{a}_{1}=8[\/latex], and we are given that [latex]n=11[\/latex].We can find [latex]r[\/latex] by dividing the second term of the series by the first.\n[latex]r=\\dfrac{-4}{8}=-\\frac{1}{2}[\/latex]\nSubstitute values for [latex]{a}_{1}, r, \\text{and} n[\/latex] into the formula and simplify.\n[latex]\\begin{align}\\\\ &amp;{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &amp;{S}_{11}=\\dfrac{8\\left(1-{\\left(-\\frac{1}{2}\\right)}^{11}\\right)}{1-\\left(-\\frac{1}{2}\\right)}\\approx 5.336 \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n \t<li>Find [latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.\n[latex]{a}_{1}=3\\cdot {2}^{1}=6[\/latex]\nWe can see from the given explicit formula that [latex]r=2[\/latex]. The upper limit of summation is 6, so [latex]n=6[\/latex].Substitute values for [latex]{a}_{1},r[\/latex], and [latex]n[\/latex] into the formula, and simplify.\n[latex]\\begin{align}\\\\ &amp;{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &amp;{S}_{6}=\\frac{6\\left(1-{2}^{6}\\right)}{1 - 2}=378 \\end{align}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nUse the formula to find the indicated partial sum of each geometric series.\n[latex]{S}_{20}[\/latex] for the series [latex]1\\text{,}000 + 500 + 250 + \\dots [\/latex]\n\n[reveal-answer q=\"922435\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"922435\"]\n\n[latex]\\approx 2,000.00[\/latex]\n\n[\/hidden-answer]\n\n&nbsp;\n\nUse the formula to determine the sum&nbsp;[latex]\\sum\\limits _{k=1}^{8}{3}^{k}[\/latex]\n\n[reveal-answer q=\"15208\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"15208\"]\n\n9,840\n\n[\/hidden-answer]\n\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19446&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Application Problem with a Geometric Series<\/h3>\nAt 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.\n\n[reveal-answer q=\"636578\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"636578\"]\n\nThe problem can be represented by a geometric series with [latex]{a}_{1}=26,750[\/latex]; [latex]n=5[\/latex]; and [latex]r=1.016[\/latex]. Substitute values for [latex]{a}_{1}[\/latex], [latex]r[\/latex], and [latex]n[\/latex] into the formula and simplify to find the total amount earned at the end of 5 years.\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&amp;=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\ {S}_{5}&amp;=\\dfrac{26\\text{,}750\\left(1-{1.016}^{5}\\right)}{1 - 1.016}\\approx 138\\text{,}099.03 \\end{align}[\/latex]<\/p>\nHe will have earned a total of $138,099.03 by the end of 5 years.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAt 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?\n\n[reveal-answer q=\"890801\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"890801\"]\n\n$275,513.31\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5865&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\n\n<\/div>\n<h2>Using the Formula for the Sum of an Infinite Geometric Series<\/h2>\nThus 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&nbsp;<em>n<\/em> terms. An&nbsp;<strong>infinite series<\/strong>&nbsp;is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\\dots[\/latex].\n\nThis series can also be written in summation notation as [latex] \\sum\\limits _{k=1}^{\\infty} 2k[\/latex],&nbsp;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&nbsp;<strong>diverges<\/strong>.\n<h3>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h3>\nIf the terms of an&nbsp;<span class=\"no-emphasis\">infinite geometric series<\/span>&nbsp;approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:\n<p style=\"text-align: center;\">[latex]1+0.2+0.04+0.008+0.0016+\\dots[\/latex]<\/p>\nThe common ratio is [latex]r=0.2[\/latex]. As&nbsp;<em>n<\/em> gets large, the values of of [latex]r^n[\/latex] get very small and approach 0.&nbsp;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]&nbsp;approach 0; the sum of a geometric series is defined when&nbsp;[latex]-1&lt;r&lt;1[\/latex].\n<div class=\"textbox\">\n<h3>DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED<\/h3>\nThe sum of an infinite series is defined if the series is geometric and&nbsp;[latex]-1&lt;r&lt;1[\/latex].\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: <strong>Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong><\/h3>\n<ol>\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,&nbsp;<em>r<\/em>, was found in step 3, check to see if&nbsp;[latex]-1&lt;r&lt;1[\/latex].&nbsp;If so, the sum is defined. If not, the sum is not defined.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Determining Whether the Sum of an Infinite Series is Defined<\/h3>\nDetermine whether the sum of each infinite series is defined.\n<ol>\n \t<li>[latex]12+8+4+\/dots[\/latex]<\/li>\n \t<li>[latex]\\dfrac{3}{4}+\\dfrac{1}{2}+\\dfrac{1}{3}+\\dots[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}{27}\\cdot\\left(\\dfrac{1}{3}\\right)^k[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}{5k}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"250515\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"250515\"]\n<ol>\n \t<li>The ratio of the second term to the first is [latex]\\frac{2}{3}[\/latex], which is not the same as the ratio of the third term to the second, [latex]\\frac{1}{2}[\/latex].&nbsp;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}[\/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}[\/latex]; the series is geometric with a common ratio of&nbsp;[latex]\\frac{1}{3}[\/latex]. The sum of the infinite series is defined.<\/li>\n \t<li>The given formula is not exponential. The series is arithmetic, not geometric and so cannot yield a finite sum.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\nDetermine whether the sum of the infinite series is defined.\n<ol>\n \t<li>[latex]\\dfrac{1}{3}+\\dfrac{1}{2}+\\dfrac{3}{4}+\\dfrac{9}{8}+\\cdots[\/latex]<\/li>\n \t<li>[latex]24+(-12)+6+(-3)+\\dots[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty} 15\\cdot(-0.3)^k[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"559520\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"559520\"]\n<ol>\n \t<li>The series is geometric, but [latex]r=\\dfrac{3}{2}&gt;1[\/latex]. The sum is not defined.<\/li>\n \t<li>The series is geometric with [latex]r=-\\dfrac{1}{2}[\/latex]. The sum is defined.<\/li>\n \t<li>The series is geometric with [latex]r=-0.3[\/latex]. The sum is defined.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<h2>Finding Sums of Infinite Series<\/h2>\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 <em>n<\/em>&nbsp;terms of a geometric series.<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex]<\/p>\nWe will examine an infinite series with [latex]r=\\frac{1}{2}[\/latex]. What happens to [latex]r^n[\/latex] as&nbsp;<em>n<\/em> increases?\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{\\left(\\frac{1}{2}\\right)}^{2} = \\frac{1}{4} \\\\&amp;{\\left(\\frac{1}{2}\\right)}^{3} = \\frac{1}{8} \\\\&amp;{\\left(\\frac{1}{2}\\right)}^{4} = \\frac{1}{16} \\end{align}[\/latex]<\/p>\nThe value of [latex]r^n[\/latex] decreases rapidly. What happens for greater values of&nbsp;<em>n<\/em>?\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{\\left(\\frac{1}{2}\\right)}^{10} = \\frac{1}{1\\text{,}024} \\\\&amp;{\\left(\\frac{1}{2}\\right)}^{20} = \\frac{1}{1\\text{,}048\\text{,}576} \\\\&amp;{\\left(\\frac{1}{2}\\right)}^{30} = \\frac{1}{1\\text{,}073\\text{,}741\\text{,}824} \\end{align}[\/latex]<\/p>\nAs&nbsp;<em>n<\/em> gets large, [latex]r^n[\/latex] gets very small. We say that as&nbsp;<em>n<\/em> increases without bound,&nbsp;[latex]r^n[\/latex] approaches 0. As&nbsp;[latex]r^n[\/latex] approaches 0,&nbsp;[latex]1-r^n[\/latex] approaches 1. When this happens the numerator approaches [latex]a_1[\/latex]. This gives us the formula for the sum of an infinite geometric series.\n<div class=\"textbox\">\n<h3>A General Note: FORMULA FOR THE SUM OF AN INFINITE GEOMETRIC SERIES<\/h3>\nThe formula for the sum of an infinite geometric series with [latex]-1&lt;r&lt;1[\/latex] is:\n<p style=\"text-align: center;\">[latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: <strong>Given an infinite geometric series, find its sum.<\/strong><\/h3>\n<ol>\n \t<li>Identify [latex]a_1[\/latex] and&nbsp;<em>r<\/em>.<\/li>\n \t<li>Confirm that [latex]-1&lt;r&lt;1[\/latex].<\/li>\n \t<li>Substitute values for&nbsp;[latex]a_1[\/latex] and&nbsp;<em>r<\/em> into the formula,&nbsp;[latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex].<\/li>\n \t<li>Simplify to find&nbsp;<em>S<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Finding the Sum of an Infinite Geometric Series<\/h3>\nFind the sum, if it exists, for the following:\n<ol>\n \t<li>[latex]10+9+8+7+\\dots[\/latex]<\/li>\n \t<li><span id=\"MJXp-Span-5192\" class=\"MJXp-mo\">[latex]248.6+99.44+39.776+\\dots[\/latex]<\/span><\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}4\\text{,}374\\cdot\\left(-\\dfrac{1}{3}\\right)^{k-1}[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\dfrac{1}{9}\\cdot\\left(\\dfrac{4}{3}\\right)^{k}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"18513\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"18513\"]\n<ol>\n \t<li>There is not a constant ratio; the series is not geometric.<\/li>\n \t<li>There is a constant ratio; the series is geometric. [latex]a_1=248.6[\/latex] and [latex]r=\\dfrac{99.44}{248.6}=0.4[\/latex], so the sum exists. Substitute&nbsp;[latex]a_1=248.6[\/latex] and [latex]r=0.4[\/latex] into the formula and simplify to find the sum.\n[latex]\\begin{align} \\\\ &amp;S=\\frac{a_1}{1-r} \\\\[1.5mm] &amp;S=\\frac{248.6}{1-0.4}=\\frac{1243}{3} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n \t<li>The formula is exponential, so the series is geometric with [latex]r=-\\frac{1}{3}[\/latex]. Find [latex]a_1[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.\n[latex]\\begin{align} \\\\ a_1=4\\text{,}374\\cdot\\left(-\\frac{1}{3}\\right)^{1-1}=4\\text{,}374 \\\\ \\text{ }\\end{align}[\/latex]\nSubstitute [latex]4\\text{,}374[\/latex] and [latex]r=-\\frac{1}{3}[\/latex] into the formula, and simplify to find the sum.\n[latex]\\begin{align}\\\\&amp;S=\\frac{a_1}{1-r} \\\\[1.5mm] &amp;S=\\frac{4\\text{,}374}{1-\\left(-\\frac{1}{3}\\right)}=3\\text{,}280.5 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n \t<li>The formula is exponential, so the series is geometric, but [latex]r&gt;1[\/latex].&nbsp;The sum does not exist.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding an Equivalent Fraction for a Repeating Decimal<\/h3>\nFind the equivalent fraction for the repeating decimal [latex]0.\\overline{3}[\/latex].\n\n[reveal-answer q=\"624358\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"624358\"]\n\nWe notice the repeating decimal [latex]0.\\overline{3}=0.333\\dots[\/latex].&nbsp;so we can rewrite the repeating decimal as a sum of terms.\n<p style=\"text-align: center;\">[latex]0.\\overline{3}=0.3+0.03+0.003+\\dots[\/latex]<\/p>\nLooking 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.\n<p style=\"text-align: center;\">[latex]\\begin{align}0.\\overline{3}&amp;=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.01)+0.3\\cdot(0.001)+\\dots \\\\ &amp;=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.1)^2+0.3\\cdot(0.1)^3+\\dots[\/latex]<\/p>\nNotice 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\n<p style=\"text-align: center;\">[latex]S=\\dfrac{a_1}{1-r} =\\dfrac{0.3}{1-0.1} =\\dfrac{0.3}{0.9} =\\dfrac{1}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\nFind the sum if it exists.\n<ol>\n \t<li>[latex]2+\\dfrac{2}{3}+\\dfrac{2}{9}+\\dots[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}{0.76k+1}[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\left(-\\dfrac{3}{8}\\right)^k[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"221023\"]Solution\/reveal-answer]\n[hidden-answer a=\"221023\"]\n<ol>\n \t<li>3<\/li>\n \t<li>The series is arithmetic. The sum does not exist.<\/li>\n \t<li>[latex]-\\dfrac{3}{11}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n[ohm_question hide_question_numbers=1]20285[\/ohm_question]\n\n[ohm_question hide_question_numbers=1]20287[\/ohm_question]\n\n<\/div>\n&nbsp;\n<h2>Annuities<\/h2>\nAt 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 <strong>annuity<\/strong> 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% <strong>annual interest<\/strong>, 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.\n\nWe 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.\n\nWe 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,[\/latex] and [latex]n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.\n<p style=\"text-align: center;\">[latex]{S}_{72}=\\dfrac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/p>\nAfter 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 [latex]72\\left(50\\right) = $3,600[\/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.\n<div class=\"textbox\">\n<h3>How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/h3>\n<ol>\n \t<li>Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\n \t<li>Determine [latex]n[\/latex], the number of deposits.<\/li>\n \t<li>Determine [latex]r[\/latex].\n<ol>\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},r,[\/latex] and [latex]n[\/latex]\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/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 class=\"textbox exercises\">\n<h3>Example: 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[reveal-answer q=\"808488\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"808488\"]\n\nThe 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.\n<p style=\"text-align: center;\">[latex]r=1+\\dfrac{0.09}{12}=1.0075[\/latex]<\/p>\nSubstitute [latex]{a}_{1}=100,r=1.0075,[\/latex] and [latex]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.\n<p style=\"text-align: center;\">[latex]{S}_{120}=\\dfrac{100\\left(1-{1.0075}^{120}\\right)}{1 - 1.0075}\\approx 19\\text{,}351.43[\/latex]<\/p>\nSo the account has $19,351.43 after the last deposit is made.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAt 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?\n\n[reveal-answer q=\"786342\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"786342\"]\n\n$92,408.18\n\n[\/hidden-answer]\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20277&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\n\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/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}=\\dfrac{{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]-1&lt;r&lt;1[\/latex]<\/td>\n<td>[latex]{S}_{n}=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n \t<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n \t<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\n \t<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\n \t<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n \t<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\n<\/ul>\n<ul>\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.<\/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.<\/li>\n \t<li>The sum of an infinite series exists if the series is geometric with [latex]-1&lt;r&lt;1[\/latex].<\/li>\n \t<li>If the sum of an infinite series exists, it can be found using a formula.<\/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.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>annuity<\/strong> an investment in which the purchaser makes a sequence of periodic, equal payments\n\n<strong>common ratio<\/strong> the ratio between any two consecutive terms in a geometric sequence\n\n<strong>diverge<\/strong> a series is said to diverge if the sum is not a real number\n\n<strong>geometric sequence<\/strong> a sequence in which the ratio of a term to a previous term is a constant\n\n<strong>geometric series<\/strong> the sum of the terms in a geometric sequence\n\n<strong>infinite series<\/strong> the sum of the terms in an infinite sequence\n\n<strong>series<\/strong> the sum of the terms in a sequence\n\n<strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series\n<h2><\/h2>\n&nbsp;\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Find the common ratio for a geometric sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">List the terms of a geometric sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use a recursive formula for a geometric sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use an explicit formula for a geometric sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of a geometric series.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of an in\ufb01nite geometric series.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Solve annuity problems.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section we will review sequences that grow in this way.<\/p>\n<h2>Terms of Geometric Sequences<\/h2>\n<h3>Finding Common Ratios<\/h3>\n<p>The yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223627\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Definition of a Geometric Sequence<\/h3>\n<p>A <strong>geometric sequence<\/strong> is one in which any term divided by the previous term is a constant. This constant is called the <strong>common ratio<\/strong> of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/h3>\n<ol>\n<li>Divide each term by the previous term.<\/li>\n<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Common Ratios<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<ol>\n<li>[latex]1,2,4,8,16,\\dots[\/latex]<\/li>\n<li>[latex]48,12,4,2,\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706624\">Show Solution<\/span><\/p>\n<div id=\"q706624\" class=\"hidden-answer\" style=\"display: none\">\n<p>Divide each term by the previous term to determine whether a common ratio exists.<\/p>\n<ol>\n<li>[latex]\\begin{align}&\\frac{2}{1}=2 && \\frac{4}{2}=2 && \\frac{8}{4}=2 && \\frac{16}{8}=2 \\end{align}[\/latex]<br \/>\nThe sequence is geometric because there is a common ratio. The common ratio is 2.<\/li>\n<li>[latex]\\begin{align}&\\frac{12}{48}=\\frac{1}{4} && \\frac{4}{12}=\\frac{1}{3} && \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]<br \/>\nThe sequence is not geometric because there is not a common ratio.<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223630\/CNX_Precalc_Figure_11_03_0022.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/h4>\n<p><em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<br \/>\n[latex]5,10,15,20,\\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q893960\">Show Solution<\/span><\/p>\n<div id=\"q893960\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68722&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the sequence geometric? If so, find the common ratio.<\/p>\n<p>[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q870692\">Show Solution<\/span><\/p>\n<div id=\"q870692\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Writing Terms of Geometric Sequences<\/h3>\n<p>Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{1}=-2 \\\\ &{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &{a}_{4}=\\left(-32\\cdot 4\\right)=-128 \\end{align}[\/latex]<\/p>\n<p>The first four terms are [latex]\\left\\{-2,-8,-32,-128\\right\\}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/h3>\n<ol>\n<li>Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n<li>Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n<li>Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Terms of a Geometric Sequence<\/h3>\n<p>List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q650557\">Show Solution<\/span><\/p>\n<div id=\"q650557\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],&nbsp;and so on.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{1}=5 \\\\ &{a}_{2}=-2{a}_{1}=-10 \\\\ &{a}_{3}=-2{a}_{2}=20 \\\\ &{a}_{4}=-2{a}_{3}=-40 \\end{align}[\/latex]<\/p>\n<p>The first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\dfrac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q228021\">Show Solution<\/span><\/p>\n<div id=\"q228021\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156689&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Explicit Formulas for Geometric Sequences<\/h2>\n<h3>Using Explicit Formulas for Geometric Sequences<\/h3>\n<p>Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/p>\n<p>Let\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}[\/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}=18\\cdot {2}^{n - 1}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223634\/CNX_Precalc_Figure_11_03_0042.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Explicit Formula for a Geometric Sequence<\/h3>\n<p>The <em>n<\/em>th term of a geometric sequence is given by the <strong>explicit formula<\/strong>:<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n<p>Given a geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]{a}_{4}=24[\/latex], find [latex]{a}_{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602906\">Show Solution<\/span><\/p>\n<div id=\"q602906\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence can be written in terms of the initial term and the common ratio [latex]r[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3,3r,3{r}^{2},3{r}^{3},\\dots[\/latex]<\/p>\n<p>Find the common ratio using the given fourth term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}{r}^{n - 1} \\\\ &{a}_{4}=3{r}^{3} && \\text{Write the fourth term of sequence in terms of }{a}_{1}\\text{ and }r \\\\ &24=3{r}^{3} && \\text{Substitute }24\\text{ for }{a}_{4} \\\\ &8={r}^{3} && \\text{Divide} \\\\ &r=2 && \\text{Solve for the common ratio} \\end{align}[\/latex]<\/p>\n<p>Find the second term by multiplying the first term by the common ratio.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{2} & =2{a}_{1} \\\\ & =2\\left(3\\right) \\\\ & =6 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q132711\">Show Solution<\/span><\/p>\n<div id=\"q132711\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{6}=16\\text{,}384[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5856&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\n<p>Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2,10,50,250,\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q202402\">Show Solution<\/span><\/p>\n<div id=\"q202402\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{10}{2}=5[\/latex]<\/p>\n<p>The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}{r}^{\\left(n - 1\\right)} \\\\ &{a}_{n}=2\\cdot {5}^{n - 1} \\end{align}[\/latex]<\/p>\n<p>The graph of this sequence shows an exponential pattern.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223636\/CNX_Precalc_Figure_11_03_0052.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write an explicit formula for the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{-1,3,-9,27,\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q521311\">Show Solution<\/span><\/p>\n<div id=\"q521311\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29756&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we can alter the explicit formula slightly by using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Geometric Sequences<\/h3>\n<p>In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.<\/p>\n<ol>\n<li>Write a formula for the student population.<\/li>\n<li>Estimate the student population in 2020.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304368\">Show Solution<\/span><\/p>\n<div id=\"q304368\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get<br \/>\n[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/li>\n<li>We can find the number of years since 2013 by subtracting.<br \/>\n[latex]2020 - 2013=7[\/latex]<br \/>\nWe are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.<br \/>\n[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<br \/>\nThe student population will be about 374 in 2020.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Write a formula for the number of hits.<\/li>\n<li>Estimate the number of hits in 5 weeks.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q67595\">Show Solution<\/span><\/p>\n<div id=\"q67595\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]<\/li>\n<li>The number of hits will be about 333.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides a short lesson on some of the topics covered in this lesson.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Geometric Sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XHyeLKZYb2w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>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<h2>Geometric Series<\/h2>\n<p>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 <strong>geometric sequence<\/strong> is a sequence in which the ratio of any two consecutive terms is the <strong>common ratio<\/strong>, [latex]r[\/latex]. We can write the sum of the first [latex]n[\/latex] terms of a geometric series as<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1}[\/latex].<\/p>\n<p>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<p style=\"text-align: center;\">[latex]r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}[\/latex]<\/p>\n<p>Next, we subtract this equation from the original equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1} \\\\ -r{S}_{n}&=-\\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}\\right) \\\\ \\hline \\left(1-r\\right){S}_{n}&={a}_{1}-{a}_{1}{r}^{n}\\end{align}[\/latex]<\/p>\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], factor [latex]a_1[\/latex] on the right hand side and divide both sides by [latex]\\left(1-r\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Sum of the First <em>n<\/em> Terms of a Geometric Series<\/h3>\n<p>A <strong>geometric series<\/strong> 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<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a geometric series, find the sum of the first <em>n<\/em> terms.<\/h3>\n<ol>\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}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\n<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: 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>\n<li>[latex]{S}_{11}[\/latex] for the series [latex]8 + -4 + 2 + \\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^6 3\\cdot {2}^{k}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q618333\">Show Solution<\/span><\/p>\n<div id=\"q618333\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{a}_{1}=8[\/latex], and we are given that [latex]n=11[\/latex].We can find [latex]r[\/latex] by dividing the second term of the series by the first.<br \/>\n[latex]r=\\dfrac{-4}{8}=-\\frac{1}{2}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1}, r, \\text{and} n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align}\\\\ &{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &{S}_{11}=\\dfrac{8\\left(1-{\\left(-\\frac{1}{2}\\right)}^{11}\\right)}{1-\\left(-\\frac{1}{2}\\right)}\\approx 5.336 \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n<li>Find [latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.<br \/>\n[latex]{a}_{1}=3\\cdot {2}^{1}=6[\/latex]<br \/>\nWe can see from the given explicit formula that [latex]r=2[\/latex]. The upper limit of summation is 6, so [latex]n=6[\/latex].Substitute values for [latex]{a}_{1},r[\/latex], and [latex]n[\/latex] into the formula, and simplify.<br \/>\n[latex]\\begin{align}\\\\ &{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &{S}_{6}=\\frac{6\\left(1-{2}^{6}\\right)}{1 - 2}=378 \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the formula to find the indicated partial sum of each geometric series.<br \/>\n[latex]{S}_{20}[\/latex] for the series [latex]1\\text{,}000 + 500 + 250 + \\dots[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q922435\">Show Solution<\/span><\/p>\n<div id=\"q922435\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\approx 2,000.00[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use the formula to determine the sum&nbsp;[latex]\\sum\\limits _{k=1}^{8}{3}^{k}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15208\">Show Solution<\/span><\/p>\n<div id=\"q15208\" class=\"hidden-answer\" style=\"display: none\">\n<p>9,840<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19446&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Application Problem with a Geometric Series<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q636578\">Show Solution<\/span><\/p>\n<div id=\"q636578\" class=\"hidden-answer\" style=\"display: none\">\n<p>The problem can be represented by a geometric series with [latex]{a}_{1}=26,750[\/latex]; [latex]n=5[\/latex]; and [latex]r=1.016[\/latex]. Substitute values for [latex]{a}_{1}[\/latex], [latex]r[\/latex], and [latex]n[\/latex] into the formula and simplify to find the total amount earned at the end of 5 years.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\ {S}_{5}&=\\dfrac{26\\text{,}750\\left(1-{1.016}^{5}\\right)}{1 - 1.016}\\approx 138\\text{,}099.03 \\end{align}[\/latex]<\/p>\n<p>He will have earned a total of $138,099.03 by the end of 5 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q890801\">Show Solution<\/span><\/p>\n<div id=\"q890801\" class=\"hidden-answer\" style=\"display: none\">\n<p>$275,513.31<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5865&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Formula for the Sum of an Infinite Geometric Series<\/h2>\n<p>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&nbsp;<em>n<\/em> terms. An&nbsp;<strong>infinite series<\/strong>&nbsp;is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\\dots[\/latex].<\/p>\n<p>This series can also be written in summation notation as [latex]\\sum\\limits _{k=1}^{\\infty} 2k[\/latex],&nbsp;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&nbsp;<strong>diverges<\/strong>.<\/p>\n<h3>Determining Whether the Sum of an Infinite Geometric Series is Defined<\/h3>\n<p>If the terms of an&nbsp;<span class=\"no-emphasis\">infinite geometric series<\/span>&nbsp;approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:<\/p>\n<p style=\"text-align: center;\">[latex]1+0.2+0.04+0.008+0.0016+\\dots[\/latex]<\/p>\n<p>The common ratio is [latex]r=0.2[\/latex]. As&nbsp;<em>n<\/em> gets large, the values of of [latex]r^n[\/latex] get very small and approach 0.&nbsp;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]&nbsp;approach 0; the sum of a geometric series is defined when&nbsp;[latex]-1<r<1[\/latex].\n\n\n<div class=\"textbox\">\n<h3>DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED<\/h3>\n<p>The sum of an infinite series is defined if the series is geometric and&nbsp;[latex]-1<r<1[\/latex].\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: <strong>Given the first several terms of an infinite series, determine if the sum of the series exists.<\/strong><\/h3>\n<ol>\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,&nbsp;<em>r<\/em>, was found in step 3, check to see if&nbsp;[latex]-1<r<1[\/latex].&nbsp;If so, the sum is defined. If not, the sum is not defined.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Determining Whether the Sum of an Infinite Series is Defined<\/h3>\n<p>Determine whether the sum of each infinite series is defined.<\/p>\n<ol>\n<li>[latex]12+8+4+\/dots[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{4}+\\dfrac{1}{2}+\\dfrac{1}{3}+\\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}{27}\\cdot\\left(\\dfrac{1}{3}\\right)^k[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}{5k}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q250515\">Show Solution<\/span><\/p>\n<div id=\"q250515\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The ratio of the second term to the first is [latex]\\frac{2}{3}[\/latex], which is not the same as the ratio of the third term to the second, [latex]\\frac{1}{2}[\/latex].&nbsp;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}[\/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}[\/latex]; the series is geometric with a common ratio of&nbsp;[latex]\\frac{1}{3}[\/latex]. The sum of the infinite series is defined.<\/li>\n<li>The given formula is not exponential. The series is arithmetic, not geometric and so cannot yield a finite sum.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Determine whether the sum of the infinite series is defined.<\/p>\n<ol>\n<li>[latex]\\dfrac{1}{3}+\\dfrac{1}{2}+\\dfrac{3}{4}+\\dfrac{9}{8}+\\cdots[\/latex]<\/li>\n<li>[latex]24+(-12)+6+(-3)+\\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty} 15\\cdot(-0.3)^k[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q559520\">Show Solution<\/span><\/p>\n<div id=\"q559520\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The series is geometric, but [latex]r=\\dfrac{3}{2}>1[\/latex]. The sum is not defined.<\/li>\n<li>The series is geometric with [latex]r=-\\dfrac{1}{2}[\/latex]. The sum is defined.<\/li>\n<li>The series is geometric with [latex]r=-0.3[\/latex]. The sum is defined.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Finding Sums of Infinite Series<\/h2>\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 <em>n<\/em>&nbsp;terms of a geometric series.<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex]<\/p>\n<p>We will examine an infinite series with [latex]r=\\frac{1}{2}[\/latex]. What happens to [latex]r^n[\/latex] as&nbsp;<em>n<\/em> increases?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{\\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{align}[\/latex]<\/p>\n<p>The value of [latex]r^n[\/latex] decreases rapidly. What happens for greater values of&nbsp;<em>n<\/em>?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{\\left(\\frac{1}{2}\\right)}^{10} = \\frac{1}{1\\text{,}024} \\\\&{\\left(\\frac{1}{2}\\right)}^{20} = \\frac{1}{1\\text{,}048\\text{,}576} \\\\&{\\left(\\frac{1}{2}\\right)}^{30} = \\frac{1}{1\\text{,}073\\text{,}741\\text{,}824} \\end{align}[\/latex]<\/p>\n<p>As&nbsp;<em>n<\/em> gets large, [latex]r^n[\/latex] gets very small. We say that as&nbsp;<em>n<\/em> increases without bound,&nbsp;[latex]r^n[\/latex] approaches 0. As&nbsp;[latex]r^n[\/latex] approaches 0,&nbsp;[latex]1-r^n[\/latex] approaches 1. When this happens the numerator approaches [latex]a_1[\/latex]. This gives us the formula for the sum of an infinite geometric series.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: FORMULA FOR THE SUM OF AN INFINITE GEOMETRIC SERIES<\/h3>\n<p>The formula for the sum of an infinite geometric series with [latex]-1<r<1[\/latex] is:\n\n\n<p style=\"text-align: center;\">[latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: <strong>Given an infinite geometric series, find its sum.<\/strong><\/h3>\n<ol>\n<li>Identify [latex]a_1[\/latex] and&nbsp;<em>r<\/em>.<\/li>\n<li>Confirm that [latex]-1<r<1[\/latex].<\/li>\n<li>Substitute values for&nbsp;[latex]a_1[\/latex] and&nbsp;<em>r<\/em> into the formula,&nbsp;[latex]S=\\dfrac{{a}_{1}}{1-r}[\/latex].<\/li>\n<li>Simplify to find&nbsp;<em>S<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Finding the Sum of an Infinite Geometric Series<\/h3>\n<p>Find the sum, if it exists, for the following:<\/p>\n<ol>\n<li>[latex]10+9+8+7+\\dots[\/latex]<\/li>\n<li><span id=\"MJXp-Span-5192\" class=\"MJXp-mo\">[latex]248.6+99.44+39.776+\\dots[\/latex]<\/span><\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}4\\text{,}374\\cdot\\left(-\\dfrac{1}{3}\\right)^{k-1}[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\dfrac{1}{9}\\cdot\\left(\\dfrac{4}{3}\\right)^{k}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q18513\">Show Solution<\/span><\/p>\n<div id=\"q18513\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>There is not a constant ratio; the series is not geometric.<\/li>\n<li>There is a constant ratio; the series is geometric. [latex]a_1=248.6[\/latex] and [latex]r=\\dfrac{99.44}{248.6}=0.4[\/latex], so the sum exists. Substitute&nbsp;[latex]a_1=248.6[\/latex] and [latex]r=0.4[\/latex] into the formula and simplify to find the sum.<br \/>\n[latex]\\begin{align} \\\\ &S=\\frac{a_1}{1-r} \\\\[1.5mm] &S=\\frac{248.6}{1-0.4}=\\frac{1243}{3} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>The formula is exponential, so the series is geometric with [latex]r=-\\frac{1}{3}[\/latex]. Find [latex]a_1[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{align} \\\\ a_1=4\\text{,}374\\cdot\\left(-\\frac{1}{3}\\right)^{1-1}=4\\text{,}374 \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nSubstitute [latex]4\\text{,}374[\/latex] and [latex]r=-\\frac{1}{3}[\/latex] into the formula, and simplify to find the sum.<br \/>\n[latex]\\begin{align}\\\\&S=\\frac{a_1}{1-r} \\\\[1.5mm] &S=\\frac{4\\text{,}374}{1-\\left(-\\frac{1}{3}\\right)}=3\\text{,}280.5 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>The formula is exponential, so the series is geometric, but [latex]r>1[\/latex].&nbsp;The sum does not exist.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding an Equivalent Fraction for a Repeating Decimal<\/h3>\n<p>Find the equivalent fraction for the repeating decimal [latex]0.\\overline{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624358\">Show Solution<\/span><\/p>\n<div id=\"q624358\" class=\"hidden-answer\" style=\"display: none\">\n<p>We notice the repeating decimal [latex]0.\\overline{3}=0.333\\dots[\/latex].&nbsp;so we can rewrite the repeating decimal as a sum of terms.<\/p>\n<p style=\"text-align: center;\">[latex]0.\\overline{3}=0.3+0.03+0.003+\\dots[\/latex]<\/p>\n<p>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 style=\"text-align: center;\">[latex]\\begin{align}0.\\overline{3}&=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.01)+0.3\\cdot(0.001)+\\dots \\\\ &=0.3+0.3\\cdot(0.1)+0.3\\cdot(0.1)^2+0.3\\cdot(0.1)^3+\\dots[\/latex]<\/p>\n<p>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<p style=\"text-align: center;\">[latex]S=\\dfrac{a_1}{1-r} =\\dfrac{0.3}{1-0.1} =\\dfrac{0.3}{0.9} =\\dfrac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Find the sum if it exists.<\/p>\n<ol>\n<li>[latex]2+\\dfrac{2}{3}+\\dfrac{2}{9}+\\dots[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}{0.76k+1}[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{\\infty}\\left(-\\dfrac{3}{8}\\right)^k[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221023\">Solution\/reveal-answer]<\/p>\n<div id=\"q221023\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>3<\/li>\n<li>The series is arithmetic. The sum does not exist.<\/li>\n<li>[latex]-\\dfrac{3}{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm20285\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20285&theme=oea&iframe_resize_id=ohm20285\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm20287\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20287&theme=oea&iframe_resize_id=ohm20287\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Annuities<\/h2>\n<p>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 <strong>annuity<\/strong> 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% <strong>annual interest<\/strong>, 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>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>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,[\/latex] and [latex]n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{72}=\\dfrac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/p>\n<p>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 [latex]72\\left(50\\right) = $3,600[\/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/h3>\n<ol>\n<li>Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\n<li>Determine [latex]n[\/latex], the number of deposits.<\/li>\n<li>Determine [latex]r[\/latex].\n<ol>\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},r,[\/latex] and [latex]n[\/latex]<br \/>\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/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 class=\"textbox exercises\">\n<h3>Example: 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q808488\"><\/span>Show Solution<\/span><\/p>\n<div id=\"q808488\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p style=\"text-align: center;\">[latex]r=1+\\dfrac{0.09}{12}=1.0075[\/latex]<\/p>\n<p>Substitute [latex]{a}_{1}=100,r=1.0075,[\/latex] and [latex]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<p style=\"text-align: center;\">[latex]{S}_{120}=\\dfrac{100\\left(1-{1.0075}^{120}\\right)}{1 - 1.0075}\\approx 19\\text{,}351.43[\/latex]<\/p>\n<p>So the account has $19,351.43 after the last deposit is made.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q786342\">Show Solution<\/span><\/p>\n<div id=\"q786342\" class=\"hidden-answer\" style=\"display: none\">\n<p>$92,408.18<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20277&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}=r{a}_{n - 1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/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}=\\dfrac{{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]-1<r<1[\/latex]<\/td>\n<td>[latex]{S}_{n}=\\dfrac{{a}_{1}}{1-r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n<li>The common ratio can be found by dividing any term in the sequence by the previous term.<\/li>\n<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/li>\n<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[\/latex] for [latex]n\\ge 2[\/latex] .<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[\/latex].<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex].<\/li>\n<\/ul>\n<ul>\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.<\/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.<\/li>\n<li>The sum of an infinite series exists if the series is geometric with [latex]-1<r<1[\/latex].<\/li>\n<li>If the sum of an infinite series exists, it can be found using a formula.<\/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.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>annuity<\/strong> an investment in which the purchaser makes a sequence of periodic, equal payments<\/p>\n<p><strong>common ratio<\/strong> the ratio between any two consecutive terms in a geometric sequence<\/p>\n<p><strong>diverge<\/strong> a series is said to diverge if the sum is not a real number<\/p>\n<p><strong>geometric sequence<\/strong> a sequence in which the ratio of a term to a previous term is a constant<\/p>\n<p><strong>geometric series<\/strong> the sum of the terms in a geometric sequence<\/p>\n<p><strong>infinite series<\/strong> the sum of the terms in an infinite sequence<\/p>\n<p><strong>series<\/strong> the sum of the terms in a sequence<\/p>\n<p><strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/p>\n<h2><\/h2>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1861\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 68722. <strong>Authored by<\/strong>: Shahbazian, Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex: Determine if a Sequence is Arithmetic or Geometric (geometric). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Z9OwCMeohnE\">https:\/\/youtu.be\/Z9OwCMeohnE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 5856. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Geometric Sequences . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 29756. <strong>Authored by<\/strong>: McClure,Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li><strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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