{"id":5387,"date":"2024-10-15T18:56:42","date_gmt":"2024-10-15T18:56:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=5387"},"modified":"2024-11-21T22:41:49","modified_gmt":"2024-11-21T22:41:49","slug":"sequences-and-series-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/sequences-and-series-get-stronger\/","title":{"raw":"Sequences and Series: Get Stronger","rendered":"Sequences and Series: Get Stronger"},"content":{"raw":"<h2>Sequences and Their Notations<\/h2>\r\n<p id=\"fs-id1165137849158\">For the following exercises, write the first four terms of the sequence.<\/p>\r\n\r\n<div id=\"fs-id1165137849161\" data-type=\"exercise\"><header><\/header>\r\n<ol>\r\n \t<li>[latex]a_n = -\\frac{16}{n+1}[\/latex]<\/li>\r\n \t<li>[latex]a_n = \\frac{2^n}{n^3}[\/latex]<\/li>\r\n \t<li>[latex]a_n = 1.25 \\cdot (-4)^{n-1}[\/latex]<\/li>\r\n \t<li>[latex]a_n = \\frac{n^2}{2n+1}[\/latex]<\/li>\r\n \t<li>[latex]a_n = -\\left(\\frac{4 \\cdot (-5)^{n-1}}{5}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135404267\">For the following exercises, write the first eight terms of the piecewise sequence.<\/p>\r\n\r\n<ol start=\"6\">\r\n \t<li>[latex]a_n = \\begin{cases} \\frac{n^2}{2n+1} &amp; \\text{if } n \\leq 5 \\ n^2 - 5 &amp; \\text{if } n &gt; 5 \\end{cases}[\/latex]<\/li>\r\n \t<li>[latex]a_n = \\begin{cases} -0.6 \\cdot 5^{n-1} &amp; \\text{if } n \\text{ is prime or } 1 \\ 2.5 \\cdot (-2)^{n-1} &amp; \\text{if } n \\text{ is composite} \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135194285\">For the following exercises, write an explicit formula for each sequence.<\/p>\r\n\r\n<ol start=\"8\">\r\n \t<li>[latex]4, \\ 7, \\ 12, \\ 19, \\ 28, \\ \\dots[\/latex]<\/li>\r\n \t<li>[latex]1, \\ 1, \\ \\frac{4}{3}, \\ 2, \\ \\frac{16}{5}, \\ \\dots[\/latex]<\/li>\r\n \t<li>[latex]1, \\ -\\frac{1}{2}, \\ \\frac{1}{4}, \\ -\\frac{1}{8}, \\ \\frac{1}{16}, \\ \\dots[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137837870\">For the following exercises, write the first five terms of the sequence.<\/p>\r\n\r\n<ol start=\"11\">\r\n \t<li id=\"fs-id1165137837876\" data-type=\"problem\">[latex]a_1 = 3, \\quad a_n = (-3) , a_{n-1}[\/latex]<\/li>\r\n \t<li data-type=\"problem\">[latex]a_1 = -1, \\quad a_n = \\frac{(-3)^{n-1}}{a_{n-1} - 2}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135241197\">For the following exercises, write the first eight terms of the sequence.<\/p>\r\n\r\n<ol start=\"13\">\r\n \t<li>[latex]a_1 = \\frac{1}{24}, \\quad a_2 = 1, \\quad a_n = (2 a_{n-2})(3 a_{n-1})[\/latex]<\/li>\r\n \t<li>[latex]a_1 = 2, \\quad a_2 = 10, \\quad a_n = \\frac{2(a_{n-1} + 2)}{a_{n-2}}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137812246\">For the following exercises, write a recursive formula for each sequence.<\/p>\r\n\r\n<ol start=\"15\">\r\n \t<li>[latex]-8, \\ -6, \\ -3, \\ 1, \\ 6, \\ \\dots[\/latex]<\/li>\r\n \t<li>[latex]35, \\ 38, \\ 41, \\ 44, \\ 47, \\ \\dots[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137749574\">For the following exercises, evaluate the factorial.<\/p>\r\n\r\n<ol start=\"17\">\r\n \t<li>[latex]6![\/latex]<\/li>\r\n \t<li>[latex]\\frac{12!}{6!}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137628680\">For the following exercises, write the first four terms of the sequence.<\/p>\r\n\r\n<ol start=\"19\">\r\n \t<li>[latex]a_n = \\frac{n!}{n^2}[\/latex]<\/li>\r\n \t<li>[latex]a_n = \\frac{n!}{n^2 - n - 1}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137526803\">For the following exercises, graph the first five terms of the indicated sequence<\/p>\r\n\r\n<ol start=\"21\">\r\n \t<li>[latex]a_n = \\frac{(-1)^n}{n} + n[\/latex]<\/li>\r\n \t<li>[latex]a_1 = 2, \\quad a_n = (-a_{n-1} + 1)^2[\/latex]<\/li>\r\n \t<li>[latex]a_n = \\frac{(n+1)!}{(n-1)!}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137612033\">For the following exercise, write an explicit formula for the sequence using the first five points shown on the graph.<\/p>\r\n\r\n<ol start=\"24\">\r\n \t<li id=\"fs-id1165137612039\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-5979\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 0.5), (2, 1), (3, 2), (4, 4), and (5, 8). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"487\" height=\"315\" \/><\/li>\r\n<\/ol>\r\nFor the following exercise, write a recursive formula for the sequence using the first five points shown on the graph.\r\n<ol start=\"25\">\r\n \t<li><img class=\"alignnone size-full wp-image-5980\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170659\/84d609ae1450ddc622082c6fac791dee0533a469.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"199\" height=\"340\" \/><\/li>\r\n<\/ol>\r\n<h2>Arithmetic Sequences<\/h2>\r\n<p id=\"fs-id1165134069164\">For the following exercise, find the common difference for the arithmetic sequence provided.<\/p>\r\n\r\n<ol>\r\n \t<li>[latex]{0, \\ \\frac{1}{2}, \\ 1, \\ \\frac{3}{2}, \\ 2, \\ \\dots }[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137611113\">For the following exercise, determine whether the sequence is arithmetic. If so find the common difference.<\/p>\r\n\r\n<ol start=\"2\">\r\n \t<li>[latex]{4, \\ 16, \\ 64, \\ 256, \\ 1024, \\ \\dots }[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, write the first five terms of the arithmetic sequence given the first term and common difference.\r\n<ol start=\"3\">\r\n \t<li>[latex]a_1 = 0, \\quad d = \\frac{2}{3}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137828050\">For the following exercise, write the first five terms of the arithmetic series given two terms.<\/p>\r\n\r\n<ol start=\"4\">\r\n \t<li>[latex]a_{13} = -60, \\quad a_{33} = -160[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.\r\n<ol start=\"5\">\r\n \t<li>First term is [latex]4[\/latex], common difference is [latex]5[\/latex], find the [latex]4^{\\text{th}}[\/latex] term.<\/li>\r\n \t<li>First term is [latex]6[\/latex], common difference is [latex]7[\/latex], find the [latex]6^{\\text{th}}[\/latex] term.<\/li>\r\n<\/ol>\r\nFor the following exercises, find the first term given two terms from an arithmetic sequence.\r\n<ol start=\"7\">\r\n \t<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_6 = 12[\/latex] and [latex]a_{14} = 28[\/latex].<\/li>\r\n \t<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_8 = 40[\/latex] and [latex]a_{23} = 115[\/latex].<\/li>\r\n \t<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_{11} = 11[\/latex] and [latex]a_{21} = 16[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercise, find the specified term given two terms from an arithmetic sequence.\r\n<ol start=\"10\">\r\n \t<li>[latex]a_3 = -17.1[\/latex] and [latex]a_{10} = -15.7[\/latex]. Find [latex]a_{21}[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercise, use the recursive formula to write the first five terms of the arithmetic sequence.\r\n<ol start=\"11\">\r\n \t<li>[latex]a_1 = -19[\/latex]; [latex]a_n = a_{n-1} - 1.4[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write a recursive formula for each arithmetic sequence.\r\n<ol start=\"12\">\r\n \t<li>[latex]a = {17, \\ 26, \\ 35, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {12, \\ 17, \\ 22, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {8.9, \\ 10.3, \\ 11.7, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {\\frac{1}{5},\u00a0 \\frac{9}{20},\u00a0 \\frac{7}{10},\u00a0 \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {\\frac{1}{6},\u00a0 -\\frac{11}{12},\u00a0 -2,\u00a0 \\dots}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.\r\n<ol start=\"17\">\r\n \t<li>[latex]a = {4, \\ 11, \\ 18, \\ \\dots}[\/latex]; Find the [latex]14^{\\text{th}}[\/latex] term.<\/li>\r\n<\/ol>\r\nFor the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.\r\n<ol start=\"18\">\r\n \t<li>[latex]a_n = 24 - 4n[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write an explicit formula for each arithmetic sequence.\r\n<ol start=\"19\">\r\n \t<li>[latex]a = {3,\u00a0 5,\u00a0 7,\u00a0 \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {-5,\u00a0 95,\u00a0 195,\u00a0 \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {1.8,\u00a0 3.6,\u00a0 5.4,\u00a0 \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {15.8,\u00a0 18.5,\u00a0 21.2,\u00a0 \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a = {0,\u00a0 \\frac{1}{3},\u00a0 \\frac{2}{3},\u00a0 \\dots}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the number of terms in the given finite arithmetic sequence.\r\n<ol start=\"24\">\r\n \t<li>[latex]a = {3, \\ -4, \\ -11, \\ \\dots, \\ -60}[\/latex]<\/li>\r\n \t<li>[latex]a = {\\frac{1}{2}, \\ 2, \\ \\frac{7}{2}, \\ \\dots, \\ 8}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137659728\">For the following exercise, determine whether the graph shown represents an arithmetic sequence.<\/p>\r\n\r\n<ol start=\"26\">\r\n \t<li id=\"fs-id1165137605908\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-5981\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 1.5), (2, 2.25), (3, 3.375), (4, 5.0625), and (5, 7.5938). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"449\" height=\"623\" \/><\/li>\r\n<\/ol>\r\nFor the following exercise, use the information provided to graph the first 5 terms of the arithmetic sequence.\r\n<ol start=\"27\">\r\n \t<li>[latex]a_1 = 9; \\quad a_n = a_{n-1} - 10[\/latex]<\/li>\r\n<\/ol>\r\n<h2>Geometric Sequences<\/h2>\r\nFor the following exercise, find the common ratio for the geometric sequence.\r\n<ol>\r\n \t<li>[latex]-0.125, \\ 0.25, \\ -0.5, \\ 1, \\ -2, \\ \\dots[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the sequence is geometric. If so, find the common ratio.\r\n<ol start=\"2\">\r\n \t<li>[latex]-6, \\ -12, \\ -24, \\ -48, \\ -96, \\ \\dots[\/latex]<\/li>\r\n \t<li>[latex]-1, \\ \\frac{1}{2}, \\ -\\frac{1}{4}, \\ \\frac{1}{8}, \\ -\\frac{1}{16}, \\ \\dots[\/latex]<\/li>\r\n \t<li>[latex]0.8, \\ 4, \\ 20, \\ 100, \\ 500, \\ \\dots[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, write the first five terms of the geometric sequence, given the first term and common ratio.\r\n<ol start=\"5\">\r\n \t<li>[latex]a_1 = 5, \\quad r = \\frac{1}{5}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, write the first five terms of the geometric sequence, given any two terms.\r\n<ol start=\"6\">\r\n \t<li>[latex]a_6 = 25, \\quad a_8 = 6.25[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, find the specified term for the geometric sequence, given the first term and common ratio.\r\n<ol start=\"7\">\r\n \t<li>The first term is [latex]16[\/latex], and the common ratio is [latex]-\\frac{1}{3}[\/latex]. Find the [latex]4^{\\text{th}}[\/latex] term.<\/li>\r\n<\/ol>\r\nFor the following exercise, find the specified term for the geometric sequence, given the first four terms.\r\n<ol start=\"8\">\r\n \t<li>[latex]a_n = {-2, \\ \\frac{2}{3}, \\ -\\frac{2}{9}, \\ \\frac{2}{27}, \\ \\dots }[\/latex]. Find [latex]a_7[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercise, write the first five terms of the geometric sequence.\r\n<ol start=\"9\">\r\n \t<li>[latex]a_1 = 7, \\quad a_n = 0.2 , a_{n-1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write a recursive formula for each geometric sequence.\r\n<ol start=\"10\">\r\n \t<li>[latex]a_n = {-32, \\ -16, \\ -8, \\ -4, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {10, \\ -3, \\ 0.9, \\ -0.27, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {\\frac{3}{5}, \\ \\frac{1}{10}, \\ \\frac{1}{60}, \\ \\frac{1}{360}, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {\\frac{1}{512}, \\ -\\frac{1}{128}, \\ \\frac{1}{32}, \\ -\\frac{1}{8}, \\ \\dots}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, write the first five terms of the geometric sequence.\r\n<ol start=\"14\">\r\n \t<li>[latex]a_n = 12 \\cdot \\left(-\\frac{1}{2}\\right)^{n-1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write an explicit formula for each geometric sequence.\r\n<ol start=\"15\">\r\n \t<li>[latex]a_n = {1, \\ 3, \\ 9, \\ 27, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {0.8, \\ -4, \\ 20, \\ -100, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {-1, \\ -\\frac{4}{5}, \\ -\\frac{16}{25}, \\ -\\frac{64}{125}, \\ \\dots}[\/latex]<\/li>\r\n \t<li>[latex]a_n = {3, \\ 1, \\ \\frac{1}{3}, \\ -\\frac{1}{9}, \\ \\dots}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, find the specified term for the geometric sequence given.\r\n<ol start=\"19\">\r\n \t<li>Let [latex]a_n = -(-\\frac{1}{3})^{n-1}[\/latex]. Find [latex]a_{12}[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercises, find the number of terms in the given finite geometric sequence.\r\n<ol start=\"20\">\r\n \t<li>[latex]a_n = {2, \\ 1, \\ \\frac{1}{2}, \\ \\dots, \\ \\frac{1}{1024}}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135531584\">For the following exercise, determine whether the graph shown represents a geometric sequence.<\/p>\r\n\r\n<ol start=\"21\">\r\n \t<li id=\"fs-id1165137898910\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-5989\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"495\" height=\"509\" \/><\/li>\r\n<\/ol>\r\nFor the following exercise, use the information provided to graph the first five terms of the geometric sequence.\r\n<ol start=\"22\">\r\n \t<li>[latex]a_1 = 3, \\quad a_n = 2a_{n-1}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Series and Their Notations<\/h2>\r\nFor the following exercises, express each description of a sum using summation notation.\r\n<ol>\r\n \t<li>The sum from [latex]n = 0[\/latex] to [latex]n = 4[\/latex] of [latex]5n[\/latex]<\/li>\r\n \t<li>The sum that results from adding the number [latex]4[\/latex] five times<\/li>\r\n<\/ol>\r\nFor the following exercise, express each arithmetic sum using summation notation.\r\n<ol start=\"3\">\r\n \t<li>[latex]10 + 18 + 26 + \\dots + 162[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each arithmetic sequence.\r\n<ol start=\"4\">\r\n \t<li>[latex]\\frac{3}{2} + 2 + \\frac{5}{2} + 3 + \\frac{7}{2}[\/latex]<\/li>\r\n \t<li>[latex]3.2 + 3.4 + 3.6 + \\dots + 5.6[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, express each geometric sum using summation notation.\r\n<ol start=\"6\">\r\n \t<li>[latex]8 + 4 + 2 + \\dots + 0.125[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each geometric sequence, and then state the indicated sum.\r\n<ol start=\"7\">\r\n \t<li>[latex]9 + 3 + 1 + \\frac{1}{3} + \\frac{1}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sum_{a=1}^{11} 64 \\cdot 0.2^{a-1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.\r\n<ol start=\"9\">\r\n \t<li>[latex]2 + 1.6 + 1.28 + 1.024 + \\dots[\/latex]<\/li>\r\n \t<li>[latex]\\sum_{k=1}^{\\infty} -\\left(-\\frac{1}{2}\\right)^{k-1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the indicated sum.\r\n<ol start=\"11\">\r\n \t<li>[latex]\\sum_{n=1}^{6} n(n - 2)[\/latex]<\/li>\r\n \t<li>[latex]\\sum_{k=1}^{7} 2^k[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series to find the sum.\r\n<ol start=\"13\">\r\n \t<li>[latex]6 + \\frac{15}{2} + 9 + \\frac{21}{2} + 12 + \\frac{27}{2} + 15[\/latex]<\/li>\r\n \t<li>[latex]\\sum_{k=1}^{11} \\left(\\frac{k}{2} - \\frac{1}{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, use the formula for the sum of the first [latex]n[\/latex] terms of a geometric series to find the partial sum.\r\n<ol start=\"15\">\r\n \t<li>[latex]S_7[\/latex] for the series [latex]0.4 - 2 + 10 - 50 \\dots[\/latex]<\/li>\r\n \t<li>[latex] \\sum_{n=1}^{10} -2 \\cdot \\left(\\frac{1}{2}\\right)^{n-1} [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the sum of the infinite geometric series.\r\n<ol start=\"17\">\r\n \t<li>[latex]-1 - \\frac{1}{4} - \\frac{1}{16} - \\frac{1}{64} \\dots[\/latex]<\/li>\r\n \t<li>[latex]\\sum_{n=1}^{\\infty} 4.6 \\cdot 0.5^{n-1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.\r\n<ol start=\"19\">\r\n \t<li>Deposit amount: [latex]$150[\/latex]; total deposits: [latex]24[\/latex]; interest rate: [latex]3 \\%[\/latex], compounded monthly<\/li>\r\n \t<li>Deposit amount: [latex]$100[\/latex]; total deposits: [latex]120[\/latex]; interest rate: [latex]10 \\%[\/latex], compounded semi-annually<\/li>\r\n<\/ol>\r\nReal-World Applications\r\n<ol start=\"21\">\r\n \t<li>A boulder rolled down a mountain, traveling [latex]6[\/latex] feet in the first second. Each successive second, its distance increased by [latex]8[\/latex] feet. How far did the boulder travel after [latex]10[\/latex] seconds?<\/li>\r\n \t<li>A pendulum travels a distance of [latex]3[\/latex] feet on its first swing. On each successive swing, it travels [latex]\\frac{3}{4}[\/latex] the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?<\/li>\r\n<\/ol>","rendered":"<h2>Sequences and Their Notations<\/h2>\n<p id=\"fs-id1165137849158\">For the following exercises, write the first four terms of the sequence.<\/p>\n<div id=\"fs-id1165137849161\" data-type=\"exercise\">\n<header><\/header>\n<ol>\n<li>[latex]a_n = -\\frac{16}{n+1}[\/latex]<\/li>\n<li>[latex]a_n = \\frac{2^n}{n^3}[\/latex]<\/li>\n<li>[latex]a_n = 1.25 \\cdot (-4)^{n-1}[\/latex]<\/li>\n<li>[latex]a_n = \\frac{n^2}{2n+1}[\/latex]<\/li>\n<li>[latex]a_n = -\\left(\\frac{4 \\cdot (-5)^{n-1}}{5}\\right)[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165135404267\">For the following exercises, write the first eight terms of the piecewise sequence.<\/p>\n<ol start=\"6\">\n<li>[latex]a_n = \\begin{cases} \\frac{n^2}{2n+1} & \\text{if } n \\leq 5 \\ n^2 - 5 & \\text{if } n > 5 \\end{cases}[\/latex]<\/li>\n<li>[latex]a_n = \\begin{cases} -0.6 \\cdot 5^{n-1} & \\text{if } n \\text{ is prime or } 1 \\ 2.5 \\cdot (-2)^{n-1} & \\text{if } n \\text{ is composite} \\end{cases}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165135194285\">For the following exercises, write an explicit formula for each sequence.<\/p>\n<ol start=\"8\">\n<li>[latex]4, \\ 7, \\ 12, \\ 19, \\ 28, \\ \\dots[\/latex]<\/li>\n<li>[latex]1, \\ 1, \\ \\frac{4}{3}, \\ 2, \\ \\frac{16}{5}, \\ \\dots[\/latex]<\/li>\n<li>[latex]1, \\ -\\frac{1}{2}, \\ \\frac{1}{4}, \\ -\\frac{1}{8}, \\ \\frac{1}{16}, \\ \\dots[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137837870\">For the following exercises, write the first five terms of the sequence.<\/p>\n<ol start=\"11\">\n<li id=\"fs-id1165137837876\" data-type=\"problem\">[latex]a_1 = 3, \\quad a_n = (-3) , a_{n-1}[\/latex]<\/li>\n<li data-type=\"problem\">[latex]a_1 = -1, \\quad a_n = \\frac{(-3)^{n-1}}{a_{n-1} - 2}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165135241197\">For the following exercises, write the first eight terms of the sequence.<\/p>\n<ol start=\"13\">\n<li>[latex]a_1 = \\frac{1}{24}, \\quad a_2 = 1, \\quad a_n = (2 a_{n-2})(3 a_{n-1})[\/latex]<\/li>\n<li>[latex]a_1 = 2, \\quad a_2 = 10, \\quad a_n = \\frac{2(a_{n-1} + 2)}{a_{n-2}}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137812246\">For the following exercises, write a recursive formula for each sequence.<\/p>\n<ol start=\"15\">\n<li>[latex]-8, \\ -6, \\ -3, \\ 1, \\ 6, \\ \\dots[\/latex]<\/li>\n<li>[latex]35, \\ 38, \\ 41, \\ 44, \\ 47, \\ \\dots[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137749574\">For the following exercises, evaluate the factorial.<\/p>\n<ol start=\"17\">\n<li>[latex]6![\/latex]<\/li>\n<li>[latex]\\frac{12!}{6!}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137628680\">For the following exercises, write the first four terms of the sequence.<\/p>\n<ol start=\"19\">\n<li>[latex]a_n = \\frac{n!}{n^2}[\/latex]<\/li>\n<li>[latex]a_n = \\frac{n!}{n^2 - n - 1}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137526803\">For the following exercises, graph the first five terms of the indicated sequence<\/p>\n<ol start=\"21\">\n<li>[latex]a_n = \\frac{(-1)^n}{n} + n[\/latex]<\/li>\n<li>[latex]a_1 = 2, \\quad a_n = (-a_{n-1} + 1)^2[\/latex]<\/li>\n<li>[latex]a_n = \\frac{(n+1)!}{(n-1)!}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137612033\">For the following exercise, write an explicit formula for the sequence using the first five points shown on the graph.<\/p>\n<ol start=\"24\">\n<li id=\"fs-id1165137612039\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5979\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 0.5), (2, 1), (3, 2), (4, 4), and (5, 8). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"487\" height=\"315\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d-300x194.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d-65x42.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d-225x146.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170632\/338d79f78ddc1fb48e1dd465e24241bb423dfa1d-350x226.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<\/ol>\n<p>For the following exercise, write a recursive formula for the sequence using the first five points shown on the graph.<\/p>\n<ol start=\"25\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5980\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170659\/84d609ae1450ddc622082c6fac791dee0533a469.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"199\" height=\"340\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170659\/84d609ae1450ddc622082c6fac791dee0533a469.jpg 199w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170659\/84d609ae1450ddc622082c6fac791dee0533a469-176x300.jpg 176w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30170659\/84d609ae1450ddc622082c6fac791dee0533a469-65x111.jpg 65w\" sizes=\"(max-width: 199px) 100vw, 199px\" \/><\/li>\n<\/ol>\n<h2>Arithmetic Sequences<\/h2>\n<p id=\"fs-id1165134069164\">For the following exercise, find the common difference for the arithmetic sequence provided.<\/p>\n<ol>\n<li>[latex]{0, \\ \\frac{1}{2}, \\ 1, \\ \\frac{3}{2}, \\ 2, \\ \\dots }[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137611113\">For the following exercise, determine whether the sequence is arithmetic. If so find the common difference.<\/p>\n<ol start=\"2\">\n<li>[latex]{4, \\ 16, \\ 64, \\ 256, \\ 1024, \\ \\dots }[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, write the first five terms of the arithmetic sequence given the first term and common difference.<\/p>\n<ol start=\"3\">\n<li>[latex]a_1 = 0, \\quad d = \\frac{2}{3}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137828050\">For the following exercise, write the first five terms of the arithmetic series given two terms.<\/p>\n<ol start=\"4\">\n<li>[latex]a_{13} = -60, \\quad a_{33} = -160[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.<\/p>\n<ol start=\"5\">\n<li>First term is [latex]4[\/latex], common difference is [latex]5[\/latex], find the [latex]4^{\\text{th}}[\/latex] term.<\/li>\n<li>First term is [latex]6[\/latex], common difference is [latex]7[\/latex], find the [latex]6^{\\text{th}}[\/latex] term.<\/li>\n<\/ol>\n<p>For the following exercises, find the first term given two terms from an arithmetic sequence.<\/p>\n<ol start=\"7\">\n<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_6 = 12[\/latex] and [latex]a_{14} = 28[\/latex].<\/li>\n<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_8 = 40[\/latex] and [latex]a_{23} = 115[\/latex].<\/li>\n<li>Find the first term or [latex]a_1[\/latex] of an arithmetic sequence if [latex]a_{11} = 11[\/latex] and [latex]a_{21} = 16[\/latex].<\/li>\n<\/ol>\n<p>For the following exercise, find the specified term given two terms from an arithmetic sequence.<\/p>\n<ol start=\"10\">\n<li>[latex]a_3 = -17.1[\/latex] and [latex]a_{10} = -15.7[\/latex]. Find [latex]a_{21}[\/latex].<\/li>\n<\/ol>\n<p>For the following exercise, use the recursive formula to write the first five terms of the arithmetic sequence.<\/p>\n<ol start=\"11\">\n<li>[latex]a_1 = -19[\/latex]; [latex]a_n = a_{n-1} - 1.4[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write a recursive formula for each arithmetic sequence.<\/p>\n<ol start=\"12\">\n<li>[latex]a = {17, \\ 26, \\ 35, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a = {12, \\ 17, \\ 22, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a = {8.9, \\ 10.3, \\ 11.7, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a = {\\frac{1}{5},\u00a0 \\frac{9}{20},\u00a0 \\frac{7}{10},\u00a0 \\dots}[\/latex]<\/li>\n<li>[latex]a = {\\frac{1}{6},\u00a0 -\\frac{11}{12},\u00a0 -2,\u00a0 \\dots}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.<\/p>\n<ol start=\"17\">\n<li>[latex]a = {4, \\ 11, \\ 18, \\ \\dots}[\/latex]; Find the [latex]14^{\\text{th}}[\/latex] term.<\/li>\n<\/ol>\n<p>For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.<\/p>\n<ol start=\"18\">\n<li>[latex]a_n = 24 - 4n[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write an explicit formula for each arithmetic sequence.<\/p>\n<ol start=\"19\">\n<li>[latex]a = {3,\u00a0 5,\u00a0 7,\u00a0 \\dots}[\/latex]<\/li>\n<li>[latex]a = {-5,\u00a0 95,\u00a0 195,\u00a0 \\dots}[\/latex]<\/li>\n<li>[latex]a = {1.8,\u00a0 3.6,\u00a0 5.4,\u00a0 \\dots}[\/latex]<\/li>\n<li>[latex]a = {15.8,\u00a0 18.5,\u00a0 21.2,\u00a0 \\dots}[\/latex]<\/li>\n<li>[latex]a = {0,\u00a0 \\frac{1}{3},\u00a0 \\frac{2}{3},\u00a0 \\dots}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the number of terms in the given finite arithmetic sequence.<\/p>\n<ol start=\"24\">\n<li>[latex]a = {3, \\ -4, \\ -11, \\ \\dots, \\ -60}[\/latex]<\/li>\n<li>[latex]a = {\\frac{1}{2}, \\ 2, \\ \\frac{7}{2}, \\ \\dots, \\ 8}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165137659728\">For the following exercise, determine whether the graph shown represents an arithmetic sequence.<\/p>\n<ol start=\"26\">\n<li id=\"fs-id1165137605908\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5981\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 1.5), (2, 2.25), (3, 3.375), (4, 5.0625), and (5, 7.5938). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"449\" height=\"623\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368.jpg 449w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368-216x300.jpg 216w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368-65x90.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368-225x312.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30171317\/e767040531a0859fb344480ed0eb3df488577368-350x486.jpg 350w\" sizes=\"(max-width: 449px) 100vw, 449px\" \/><\/li>\n<\/ol>\n<p>For the following exercise, use the information provided to graph the first 5 terms of the arithmetic sequence.<\/p>\n<ol start=\"27\">\n<li>[latex]a_1 = 9; \\quad a_n = a_{n-1} - 10[\/latex]<\/li>\n<\/ol>\n<h2>Geometric Sequences<\/h2>\n<p>For the following exercise, find the common ratio for the geometric sequence.<\/p>\n<ol>\n<li>[latex]-0.125, \\ 0.25, \\ -0.5, \\ 1, \\ -2, \\ \\dots[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.<\/p>\n<ol start=\"2\">\n<li>[latex]-6, \\ -12, \\ -24, \\ -48, \\ -96, \\ \\dots[\/latex]<\/li>\n<li>[latex]-1, \\ \\frac{1}{2}, \\ -\\frac{1}{4}, \\ \\frac{1}{8}, \\ -\\frac{1}{16}, \\ \\dots[\/latex]<\/li>\n<li>[latex]0.8, \\ 4, \\ 20, \\ 100, \\ 500, \\ \\dots[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, write the first five terms of the geometric sequence, given the first term and common ratio.<\/p>\n<ol start=\"5\">\n<li>[latex]a_1 = 5, \\quad r = \\frac{1}{5}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, write the first five terms of the geometric sequence, given any two terms.<\/p>\n<ol start=\"6\">\n<li>[latex]a_6 = 25, \\quad a_8 = 6.25[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, find the specified term for the geometric sequence, given the first term and common ratio.<\/p>\n<ol start=\"7\">\n<li>The first term is [latex]16[\/latex], and the common ratio is [latex]-\\frac{1}{3}[\/latex]. Find the [latex]4^{\\text{th}}[\/latex] term.<\/li>\n<\/ol>\n<p>For the following exercise, find the specified term for the geometric sequence, given the first four terms.<\/p>\n<ol start=\"8\">\n<li>[latex]a_n = {-2, \\ \\frac{2}{3}, \\ -\\frac{2}{9}, \\ \\frac{2}{27}, \\ \\dots }[\/latex]. Find [latex]a_7[\/latex].<\/li>\n<\/ol>\n<p>For the following exercise, write the first five terms of the geometric sequence.<\/p>\n<ol start=\"9\">\n<li>[latex]a_1 = 7, \\quad a_n = 0.2 , a_{n-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write a recursive formula for each geometric sequence.<\/p>\n<ol start=\"10\">\n<li>[latex]a_n = {-32, \\ -16, \\ -8, \\ -4, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {10, \\ -3, \\ 0.9, \\ -0.27, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {\\frac{3}{5}, \\ \\frac{1}{10}, \\ \\frac{1}{60}, \\ \\frac{1}{360}, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {\\frac{1}{512}, \\ -\\frac{1}{128}, \\ \\frac{1}{32}, \\ -\\frac{1}{8}, \\ \\dots}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, write the first five terms of the geometric sequence.<\/p>\n<ol start=\"14\">\n<li>[latex]a_n = 12 \\cdot \\left(-\\frac{1}{2}\\right)^{n-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write an explicit formula for each geometric sequence.<\/p>\n<ol start=\"15\">\n<li>[latex]a_n = {1, \\ 3, \\ 9, \\ 27, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {0.8, \\ -4, \\ 20, \\ -100, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {-1, \\ -\\frac{4}{5}, \\ -\\frac{16}{25}, \\ -\\frac{64}{125}, \\ \\dots}[\/latex]<\/li>\n<li>[latex]a_n = {3, \\ 1, \\ \\frac{1}{3}, \\ -\\frac{1}{9}, \\ \\dots}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, find the specified term for the geometric sequence given.<\/p>\n<ol start=\"19\">\n<li>Let [latex]a_n = -(-\\frac{1}{3})^{n-1}[\/latex]. Find [latex]a_{12}[\/latex].<\/li>\n<\/ol>\n<p>For the following exercises, find the number of terms in the given finite geometric sequence.<\/p>\n<ol start=\"20\">\n<li>[latex]a_n = {2, \\ 1, \\ \\frac{1}{2}, \\ \\dots, \\ \\frac{1}{1024}}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1165135531584\">For the following exercise, determine whether the graph shown represents a geometric sequence.<\/p>\n<ol start=\"21\">\n<li id=\"fs-id1165137898910\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5989\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"495\" height=\"509\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e.jpg 495w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e-292x300.jpg 292w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e-65x67.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e-225x231.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30172116\/ea3b038477ac701c44adbb5ebbe07c3de8fbeb7e-350x360.jpg 350w\" sizes=\"(max-width: 495px) 100vw, 495px\" \/><\/li>\n<\/ol>\n<p>For the following exercise, use the information provided to graph the first five terms of the geometric sequence.<\/p>\n<ol start=\"22\">\n<li>[latex]a_1 = 3, \\quad a_n = 2a_{n-1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<h2>Series and Their Notations<\/h2>\n<p>For the following exercises, express each description of a sum using summation notation.<\/p>\n<ol>\n<li>The sum from [latex]n = 0[\/latex] to [latex]n = 4[\/latex] of [latex]5n[\/latex]<\/li>\n<li>The sum that results from adding the number [latex]4[\/latex] five times<\/li>\n<\/ol>\n<p>For the following exercise, express each arithmetic sum using summation notation.<\/p>\n<ol start=\"3\">\n<li>[latex]10 + 18 + 26 + \\dots + 162[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each arithmetic sequence.<\/p>\n<ol start=\"4\">\n<li>[latex]\\frac{3}{2} + 2 + \\frac{5}{2} + 3 + \\frac{7}{2}[\/latex]<\/li>\n<li>[latex]3.2 + 3.4 + 3.6 + \\dots + 5.6[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, express each geometric sum using summation notation.<\/p>\n<ol start=\"6\">\n<li>[latex]8 + 4 + 2 + \\dots + 0.125[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of each geometric sequence, and then state the indicated sum.<\/p>\n<ol start=\"7\">\n<li>[latex]9 + 3 + 1 + \\frac{1}{3} + \\frac{1}{9}[\/latex]<\/li>\n<li>[latex]\\sum_{a=1}^{11} 64 \\cdot 0.2^{a-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.<\/p>\n<ol start=\"9\">\n<li>[latex]2 + 1.6 + 1.28 + 1.024 + \\dots[\/latex]<\/li>\n<li>[latex]\\sum_{k=1}^{\\infty} -\\left(-\\frac{1}{2}\\right)^{k-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the indicated sum.<\/p>\n<ol start=\"11\">\n<li>[latex]\\sum_{n=1}^{6} n(n - 2)[\/latex]<\/li>\n<li>[latex]\\sum_{k=1}^{7} 2^k[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series to find the sum.<\/p>\n<ol start=\"13\">\n<li>[latex]6 + \\frac{15}{2} + 9 + \\frac{21}{2} + 12 + \\frac{27}{2} + 15[\/latex]<\/li>\n<li>[latex]\\sum_{k=1}^{11} \\left(\\frac{k}{2} - \\frac{1}{2}\\right)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, use the formula for the sum of the first [latex]n[\/latex] terms of a geometric series to find the partial sum.<\/p>\n<ol start=\"15\">\n<li>[latex]S_7[\/latex] for the series [latex]0.4 - 2 + 10 - 50 \\dots[\/latex]<\/li>\n<li>[latex]\\sum_{n=1}^{10} -2 \\cdot \\left(\\frac{1}{2}\\right)^{n-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the sum of the infinite geometric series.<\/p>\n<ol start=\"17\">\n<li>[latex]-1 - \\frac{1}{4} - \\frac{1}{16} - \\frac{1}{64} \\dots[\/latex]<\/li>\n<li>[latex]\\sum_{n=1}^{\\infty} 4.6 \\cdot 0.5^{n-1}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.<\/p>\n<ol start=\"19\">\n<li>Deposit amount: [latex]$150[\/latex]; total deposits: [latex]24[\/latex]; interest rate: [latex]3 \\%[\/latex], compounded monthly<\/li>\n<li>Deposit amount: [latex]$100[\/latex]; total deposits: [latex]120[\/latex]; interest rate: [latex]10 \\%[\/latex], compounded semi-annually<\/li>\n<\/ol>\n<p>Real-World Applications<\/p>\n<ol start=\"21\">\n<li>A boulder rolled down a mountain, traveling [latex]6[\/latex] feet in the first second. Each successive second, its distance increased by [latex]8[\/latex] feet. How far did the boulder travel after [latex]10[\/latex] seconds?<\/li>\n<li>A pendulum travels a distance of [latex]3[\/latex] feet on its first swing. On each successive swing, it travels [latex]\\frac{3}{4}[\/latex] the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":30,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":363,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5387"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5387\/revisions"}],"predecessor-version":[{"id":5992,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5387\/revisions\/5992"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/363"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5387\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=5387"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=5387"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=5387"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=5387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}