{"id":2025,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=2025"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"solutions-15","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solutions-15\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"<h2>Solutions to Try Its<\/h2>\n1.\u00a0The first five terms are [latex]\\left\\{1,6, 11, 16, 21\\right\\}[\/latex].\n\n2. The first five terms are [latex]\\left\\{-2, 2, -\\frac{3}{2}, 1,\\text{ }-\\frac{5}{8}\\right\\}[\/latex].\n\n3.\u00a0The first six terms are [latex]\\left\\{2,\\text{ }5,\\text{ }54,\\text{ }10,\\text{ }250,\\text{ }15\\right\\}[\/latex].\n\n4.\u00a0[latex]{a}_{n}={\\left(-1\\right)}^{n+1}{9}^{n}[\/latex]\n\n5.\u00a0[latex]{a}_{n}=-\\frac{{3}^{n}}{4n}[\/latex]\n\n6.\u00a0[latex]{a}_{n}={e}^{n - 3}[\/latex]\n\n7.\u00a0[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]\n\n8.\u00a0[latex]\\left\\{0, 1, 1, 1, 2, 3, \\frac{5}{2},\\text{ }\\frac{17}{6}\\right\\}[\/latex].\n\n9.\u00a0The first five terms are [latex]\\left\\{1, \\frac{3}{2}, 4,\\text{ }15,\\text{ }72\\right\\}[\/latex].\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1.\u00a0A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.\n\n3.\u00a0Yes, both sets go on indefinitely, so they are both infinite sequences.\n\n5.\u00a0A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out [latex]\\text{13}\\cdot \\text{12}\\cdot \\text{11}\\cdot \\text{10}\\cdot \\text{9}\\cdot \\text{8}\\cdot \\text{7}\\cdot \\text{6}\\cdot \\text{5}\\cdot \\text{4}\\cdot \\text{3}\\cdot \\text{2}\\cdot \\text{1}\\text{.}[\/latex]\n\n7.\u00a0First four terms: [latex]-8,\\text{ }-\\frac{16}{3},\\text{ }-4,\\text{ }-\\frac{16}{5}[\/latex]\n\n9.\u00a0First four terms: [latex]2,\\text{ }\\frac{1}{2},\\text{ }\\frac{8}{27},\\text{ }\\frac{1}{4}[\/latex] .\n\n11.\u00a0First four terms: [latex]1.25,\\text{ }-5,\\text{ }20,\\text{ }-80[\/latex] .\n\n13.\u00a0First four terms: [latex]\\frac{1}{3},\\text{ }\\frac{4}{5},\\text{ }\\frac{9}{7},\\text{ }\\frac{16}{9}[\/latex] .\n\n15.\u00a0First four terms: [latex]-\\frac{4}{5},\\text{ }4,\\text{ }-20,\\text{ }100[\/latex]\n\n17.\u00a0[latex]\\frac{1}{3},\\text{ }\\frac{4}{5},\\text{ }\\frac{9}{7},\\text{ }\\frac{16}{9},\\text{ }\\frac{25}{11},\\text{ }31,\\text{ }44,\\text{ }59[\/latex]\n\n19.\u00a0[latex]-0.6,-3,-15,-20,-375,-80,-9375,-320[\/latex]\n\n21.\u00a0[latex]{a}_{n}={n}^{2}+3[\/latex]\n\n23.\u00a0[latex]{a}_{n}=\\frac{{2}^{n}}{2n}\\text{ or }\\frac{{2}^{n - 1}}{n}[\/latex]\n\n25.\u00a0[latex]{a}_{n}={\\left(-\\frac{1}{2}\\right)}^{n - 1}[\/latex]\n\n27.\u00a0First five terms: [latex]3,\\text{ }-9,\\text{ }27,\\text{ }-81,\\text{ }243[\/latex]\n\n29.\u00a0First five terms: [latex]-1,\\text{ }1,\\text{ }-9,\\text{ }\\frac{27}{11},\\text{ }\\frac{891}{5}[\/latex]\n\n31.\u00a0[latex]\\frac{1}{24},\\text{ 1, }\\frac{1}{4},\\text{ }\\frac{3}{2},\\text{ }\\frac{9}{4},\\text{ }\\frac{81}{4},\\text{ }\\frac{2187}{8},\\text{ }\\frac{531,441}{16}[\/latex]\n\n33.\u00a0[latex]2,\\text{ }10,\\text{ }12,\\text{ }\\frac{14}{5},\\text{ }\\frac{4}{5},\\text{ }2,\\text{ }10,\\text{ }12[\/latex]\n\n35.\u00a0[latex]{a}_{1}=-8,{a}_{n}={a}_{n - 1}+n[\/latex]\n\n37.\u00a0[latex]{a}_{1}=35,{a}_{n}={a}_{n - 1}+3[\/latex]\n\n39.\u00a0[latex]720[\/latex]\n\n41.\u00a0[latex]665,280[\/latex]\n\n43.\u00a0First four terms: [latex]1,\\frac{1}{2},\\frac{2}{3},\\frac{3}{2}[\/latex]\n\n45.\u00a0First four terms: [latex]-1,2,\\frac{6}{5},\\frac{24}{11}[\/latex]\n\n47.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202513\/CNX_Precalc_Figure_11_01_201.jpg\" alt=\"Graph of a scattered plot with points at (1, 0), (2, 5\/2), (3, 8\/3), (4, 17\/4), and (5, 24\/5). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\"\/>\n\n49.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202515\/CNX_Precalc_Figure_11_01_203.jpg\" alt=\"Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\"\/>\n\n51.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202516\/CNX_Precalc_Figure_11_01_205.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\"\/>\n\n53.\u00a0[latex]{a}_{n}={2}^{n - 2}[\/latex]\n\n55.\u00a0[latex]{a}_{1}=6,\\text{ }{a}_{n}=2{a}_{n - 1}-5[\/latex]\n\n57.\u00a0First five terms: [latex]\\frac{29}{37},\\frac{152}{111},\\frac{716}{333},\\frac{3188}{999},\\frac{13724}{2997}[\/latex]\n\n59.\u00a0First five terms: [latex]2,3,5,17,65537[\/latex]\n\n61.\u00a0[latex]{a}_{10}=7,257,600[\/latex]\n\n63.\u00a0First six terms: [latex]0.042,0.146,0.875,2.385,4.708[\/latex]\n\n65.\u00a0First four terms: [latex]5.975,32.765,185.743,1057.25,6023.521[\/latex]\n\n67.\u00a0If [latex]{a}_{n}=-421[\/latex] is a term in the sequence, then solving the equation [latex]-421=-6 - 8n[\/latex] for [latex]n[\/latex] will yield a non-negative integer. However, if [latex]-421=-6 - 8n[\/latex], then [latex]n=51.875[\/latex] so [latex]{a}_{n}=-421[\/latex] is not a term in the sequence.\n\n69.\u00a0[latex]{a}_{1}=1,{a}_{2}=0,{a}_{n}={a}_{n - 1}-{a}_{n - 2}[\/latex]\n\n71.\u00a0[latex]\\frac{\\left(n+2\\right)!}{\\left(n - 1\\right)!}=\\frac{\\left(n+2\\right)\\cdot \\left(n+1\\right)\\cdot \\left(n\\right)\\cdot \\left(n - 1\\right)\\cdot ...\\cdot 3\\cdot 2\\cdot 1}{\\left(n - 1\\right)\\cdot ...\\cdot 3\\cdot 2\\cdot 1}=n\\left(n+1\\right)\\left(n+2\\right)={n}^{3}+3{n}^{2}+2n[\/latex]","rendered":"<h2>Solutions to Try Its<\/h2>\n<p>1.\u00a0The first five terms are [latex]\\left\\{1,6, 11, 16, 21\\right\\}[\/latex].<\/p>\n<p>2. The first five terms are [latex]\\left\\{-2, 2, -\\frac{3}{2}, 1,\\text{ }-\\frac{5}{8}\\right\\}[\/latex].<\/p>\n<p>3.\u00a0The first six terms are [latex]\\left\\{2,\\text{ }5,\\text{ }54,\\text{ }10,\\text{ }250,\\text{ }15\\right\\}[\/latex].<\/p>\n<p>4.\u00a0[latex]{a}_{n}={\\left(-1\\right)}^{n+1}{9}^{n}[\/latex]<\/p>\n<p>5.\u00a0[latex]{a}_{n}=-\\frac{{3}^{n}}{4n}[\/latex]<\/p>\n<p>6.\u00a0[latex]{a}_{n}={e}^{n - 3}[\/latex]<\/p>\n<p>7.\u00a0[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]<\/p>\n<p>8.\u00a0[latex]\\left\\{0, 1, 1, 1, 2, 3, \\frac{5}{2},\\text{ }\\frac{17}{6}\\right\\}[\/latex].<\/p>\n<p>9.\u00a0The first five terms are [latex]\\left\\{1, \\frac{3}{2}, 4,\\text{ }15,\\text{ }72\\right\\}[\/latex].<\/p>\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.\u00a0A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.<\/p>\n<p>3.\u00a0Yes, both sets go on indefinitely, so they are both infinite sequences.<\/p>\n<p>5.\u00a0A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out [latex]\\text{13}\\cdot \\text{12}\\cdot \\text{11}\\cdot \\text{10}\\cdot \\text{9}\\cdot \\text{8}\\cdot \\text{7}\\cdot \\text{6}\\cdot \\text{5}\\cdot \\text{4}\\cdot \\text{3}\\cdot \\text{2}\\cdot \\text{1}\\text{.}[\/latex]<\/p>\n<p>7.\u00a0First four terms: [latex]-8,\\text{ }-\\frac{16}{3},\\text{ }-4,\\text{ }-\\frac{16}{5}[\/latex]<\/p>\n<p>9.\u00a0First four terms: [latex]2,\\text{ }\\frac{1}{2},\\text{ }\\frac{8}{27},\\text{ }\\frac{1}{4}[\/latex] .<\/p>\n<p>11.\u00a0First four terms: [latex]1.25,\\text{ }-5,\\text{ }20,\\text{ }-80[\/latex] .<\/p>\n<p>13.\u00a0First four terms: [latex]\\frac{1}{3},\\text{ }\\frac{4}{5},\\text{ }\\frac{9}{7},\\text{ }\\frac{16}{9}[\/latex] .<\/p>\n<p>15.\u00a0First four terms: [latex]-\\frac{4}{5},\\text{ }4,\\text{ }-20,\\text{ }100[\/latex]<\/p>\n<p>17.\u00a0[latex]\\frac{1}{3},\\text{ }\\frac{4}{5},\\text{ }\\frac{9}{7},\\text{ }\\frac{16}{9},\\text{ }\\frac{25}{11},\\text{ }31,\\text{ }44,\\text{ }59[\/latex]<\/p>\n<p>19.\u00a0[latex]-0.6,-3,-15,-20,-375,-80,-9375,-320[\/latex]<\/p>\n<p>21.\u00a0[latex]{a}_{n}={n}^{2}+3[\/latex]<\/p>\n<p>23.\u00a0[latex]{a}_{n}=\\frac{{2}^{n}}{2n}\\text{ or }\\frac{{2}^{n - 1}}{n}[\/latex]<\/p>\n<p>25.\u00a0[latex]{a}_{n}={\\left(-\\frac{1}{2}\\right)}^{n - 1}[\/latex]<\/p>\n<p>27.\u00a0First five terms: [latex]3,\\text{ }-9,\\text{ }27,\\text{ }-81,\\text{ }243[\/latex]<\/p>\n<p>29.\u00a0First five terms: [latex]-1,\\text{ }1,\\text{ }-9,\\text{ }\\frac{27}{11},\\text{ }\\frac{891}{5}[\/latex]<\/p>\n<p>31.\u00a0[latex]\\frac{1}{24},\\text{ 1, }\\frac{1}{4},\\text{ }\\frac{3}{2},\\text{ }\\frac{9}{4},\\text{ }\\frac{81}{4},\\text{ }\\frac{2187}{8},\\text{ }\\frac{531,441}{16}[\/latex]<\/p>\n<p>33.\u00a0[latex]2,\\text{ }10,\\text{ }12,\\text{ }\\frac{14}{5},\\text{ }\\frac{4}{5},\\text{ }2,\\text{ }10,\\text{ }12[\/latex]<\/p>\n<p>35.\u00a0[latex]{a}_{1}=-8,{a}_{n}={a}_{n - 1}+n[\/latex]<\/p>\n<p>37.\u00a0[latex]{a}_{1}=35,{a}_{n}={a}_{n - 1}+3[\/latex]<\/p>\n<p>39.\u00a0[latex]720[\/latex]<\/p>\n<p>41.\u00a0[latex]665,280[\/latex]<\/p>\n<p>43.\u00a0First four terms: [latex]1,\\frac{1}{2},\\frac{2}{3},\\frac{3}{2}[\/latex]<\/p>\n<p>45.\u00a0First four terms: [latex]-1,2,\\frac{6}{5},\\frac{24}{11}[\/latex]<\/p>\n<p>47.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202513\/CNX_Precalc_Figure_11_01_201.jpg\" alt=\"Graph of a scattered plot with points at (1, 0), (2, 5\/2), (3, 8\/3), (4, 17\/4), and (5, 24\/5). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>49.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202515\/CNX_Precalc_Figure_11_01_203.jpg\" alt=\"Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>51.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202516\/CNX_Precalc_Figure_11_01_205.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>53.\u00a0[latex]{a}_{n}={2}^{n - 2}[\/latex]<\/p>\n<p>55.\u00a0[latex]{a}_{1}=6,\\text{ }{a}_{n}=2{a}_{n - 1}-5[\/latex]<\/p>\n<p>57.\u00a0First five terms: [latex]\\frac{29}{37},\\frac{152}{111},\\frac{716}{333},\\frac{3188}{999},\\frac{13724}{2997}[\/latex]<\/p>\n<p>59.\u00a0First five terms: [latex]2,3,5,17,65537[\/latex]<\/p>\n<p>61.\u00a0[latex]{a}_{10}=7,257,600[\/latex]<\/p>\n<p>63.\u00a0First six terms: [latex]0.042,0.146,0.875,2.385,4.708[\/latex]<\/p>\n<p>65.\u00a0First four terms: [latex]5.975,32.765,185.743,1057.25,6023.521[\/latex]<\/p>\n<p>67.\u00a0If [latex]{a}_{n}=-421[\/latex] is a term in the sequence, then solving the equation [latex]-421=-6 - 8n[\/latex] for [latex]n[\/latex] will yield a non-negative integer. However, if [latex]-421=-6 - 8n[\/latex], then [latex]n=51.875[\/latex] so [latex]{a}_{n}=-421[\/latex] is not a term in the sequence.<\/p>\n<p>69.\u00a0[latex]{a}_{1}=1,{a}_{2}=0,{a}_{n}={a}_{n - 1}-{a}_{n - 2}[\/latex]<\/p>\n<p>71.\u00a0[latex]\\frac{\\left(n+2\\right)!}{\\left(n - 1\\right)!}=\\frac{\\left(n+2\\right)\\cdot \\left(n+1\\right)\\cdot \\left(n\\right)\\cdot \\left(n - 1\\right)\\cdot ...\\cdot 3\\cdot 2\\cdot 1}{\\left(n - 1\\right)\\cdot ...\\cdot 3\\cdot 2\\cdot 1}=n\\left(n+1\\right)\\left(n+2\\right)={n}^{3}+3{n}^{2}+2n[\/latex]<\/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-2025\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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":276,"menu_order":8,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2025","chapter","type-chapter","status-publish","hentry"],"part":2000,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2025","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2025\/revisions"}],"predecessor-version":[{"id":2180,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2025\/revisions\/2180"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2000"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2025\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=2025"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2025"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2025"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=2025"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}