{"id":1481,"date":"2025-07-25T02:09:00","date_gmt":"2025-07-25T02:09:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1481"},"modified":"2026-03-24T07:14:32","modified_gmt":"2026-03-24T07:14:32","slug":"sequences-and-their-notations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sequences-and-their-notations-fresh-take\/","title":{"raw":"Sequences and Their Notations: Fresh Take","rendered":"Sequences and Their Notations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Expand sequences defined by explicit and recursive formulas<\/li>\r\n \t<li>Recognize the notation and terms used to represent sequences<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sequences Defined by an Explicit Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of a Sequence:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A function whose domain is a subset of positive integers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each number in the sequence is called a term<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Sequences:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Finite: Has a specific number of terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite: Continues indefinitely (uses ellipsis ...)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Explicit Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Defines the nth term using its position in the sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Typically written as [latex]a_n = [\\text{ expression involving }n][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Representation Methods:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">List of terms: [latex]{a_1, a_2, a_3,...,a_n,...}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Table of values<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph (discrete points)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Piecewise Sequences:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Different formulas for different ranges of [latex]n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the <strong>explicit formula<\/strong> [latex]{t}_{n}=5n - 4[\/latex].[reveal-answer q=\"25802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"25802\"]The first five terms are [latex]\\left\\{1,6, 11, 16, 21\\right\\}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the first six terms of the sequence.\r\n<p style=\"text-align: center;\">[latex]{a_{n}}=\\begin{cases}2n^{3} &amp; \\text{if }n\\text{ is odd} \\\\[1mm] \\dfrac{5n}{2} &amp; \\text{if }n\\text{ is even}\\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"493717\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"493717\"]\r\n\r\nThe first six terms are [latex]\\left\\{2,5,54,10,250,15\\right\\}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Finding an Explicit Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Pattern Recognition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Look for relationships between terms and their positions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyze numerators and denominators separately for fraction sequences<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify patterns in signs, exponents, or bases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Alternating Sequences:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use [latex](-1)^n[\/latex] or [latex](-1)^{n-1}[\/latex] to represent alternating signs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Don't rearrange terms in numerical order<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Structure:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n = [\\text{ expression involving }n][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May involve algebraic, exponential, or trigonometric functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Always test your formula for the first few terms of the sequence<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.\r\n<p style=\"text-align: center;\">[latex]\\{9;\u221281,729;\u22126,561;59,049\\}[\/latex]<\/p>\r\n[reveal-answer q=\"806943\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"806943\"]\r\n\r\n[latex]{a}_{n}={\\left(-1\\right)}^{n+1}{9}^{n}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.\r\n<div style=\"text-align: center;\">[latex]\\left\\{-\\dfrac{3}{4},-\\dfrac{9}{8},-\\dfrac{27}{12},-\\dfrac{81}{16},-\\dfrac{243}{20},\\dots\\right\\}[\/latex]<\/div>\r\n[reveal-answer q=\"282988\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"282988\"]\r\n\r\n[latex]{a}_{n}=-\\dfrac{{3}^{n}}{4n}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.\r\n<div style=\"text-align: center;\">[latex]\\left\\{\\dfrac{1}{{e}^{2}}, \\dfrac{1}{e}, 1, e, {e}^{2},...\\right\\}[\/latex]<\/div>\r\n[reveal-answer q=\"804020\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"804020\"]\r\n\r\n[latex]{a}_{n}={e}^{n - 3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence:\r\n<div style=\"text-align: center;\">[latex]{a}_{n}=\\dfrac{4n}{{\\left(-2\\right)}^{n}}[\/latex]<\/div>\r\n<div><\/div>\r\n[reveal-answer q=\"196840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"196840\"]\r\n\r\nThe first five terms are [latex]\\left\\{-2, 2, -\\dfrac{3}{2}, 1,-\\dfrac{5}{8}\\right\\}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehhebhaa-f02dpeOaXao\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/f02dpeOaXao?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ehhebhaa-f02dpeOaXao\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780923&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ehhebhaa-f02dpeOaXao&amp;vembed=0&amp;video_id=f02dpeOaXao&amp;video_target=tpm-plugin-ehhebhaa-f02dpeOaXao\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+the+formula+of+alternating+signs+of+a+sequence_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the formula of alternating signs of a sequence\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Sequences Defined by a Recursive Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Recursive Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Defines each term using preceding term(s)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Must state initial term(s)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Structure:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Initial condition(s): [latex]a_1 = [\\text{value}][\/latex], ([latex]a_2 = [\\text{value}][\/latex], if needed)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Recursive rule: [latex]a_n = [\\text{expression involving }a_{n-1}, a_{n-2},\\text{ etc.}][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Useful for sequences defined by step-by-step processes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effective for modeling natural growth and decay phenomena<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Comparison with Explicit Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Recursive: Defines terms relative to previous terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Explicit: Defines terms directly based on their position<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Limitations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">May be computationally intensive for large [latex]n[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not always easy to find a specific term without calculating all preceding terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the recursive formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{1}&amp;=2\\\\ {a}_{n}&amp;=2{a}_{n - 1}+1\\text{, for }n\\ge 2\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"378600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"378600\"]\r\n\r\n[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebhcdfhg-RjsyEWDEQe0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RjsyEWDEQe0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebhcdfhg-RjsyEWDEQe0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851333&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebhcdfhg-RjsyEWDEQe0&amp;vembed=0&amp;video_id=RjsyEWDEQe0&amp;video_target=tpm-plugin-ebhcdfhg-RjsyEWDEQe0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Finding+Terms+in+a+Sequence+Given+a+Recursive+Formula_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Finding Terms in a Sequence Given a Recursive Formula\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Factorial Notation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Factorial:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]n! = n \\times (n-1) \\times (n-2) \\times ... \\times 3 \\times 2 \\times 1[\/latex], for [latex]n \\geq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0![\/latex] is defined as [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factorials grow very quickly<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often used in combinatorics and probability<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Application in Sequences:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can appear in both explicit and recursive formulas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often leads to rapidly converging or diverging sequences<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{\\left(n+1\\right)!}{2n}[\/latex].\r\n[reveal-answer q=\"953785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"953785\"]The first five terms are [latex]\\displaystyle \\left\\{1, \\frac{3}{2}, 4,15,72\\right\\}[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Expand sequences defined by explicit and recursive formulas<\/li>\n<li>Recognize the notation and terms used to represent sequences<\/li>\n<\/ul>\n<\/section>\n<h2>Sequences Defined by an Explicit Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of a Sequence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A function whose domain is a subset of positive integers<\/li>\n<li class=\"whitespace-normal break-words\">Each number in the sequence is called a term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Sequences:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Finite: Has a specific number of terms<\/li>\n<li class=\"whitespace-normal break-words\">Infinite: Continues indefinitely (uses ellipsis &#8230;)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Explicit Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defines the nth term using its position in the sequence<\/li>\n<li class=\"whitespace-normal break-words\">Typically written as [latex]a_n = [\\text{ expression involving }n][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Representation Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">List of terms: [latex]{a_1, a_2, a_3,...,a_n,...}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Table of values<\/li>\n<li class=\"whitespace-normal break-words\">Graph (discrete points)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Piecewise Sequences:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Different formulas for different ranges of [latex]n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the <strong>explicit formula<\/strong> [latex]{t}_{n}=5n - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q25802\">Show Solution<\/button><\/p>\n<div id=\"q25802\" class=\"hidden-answer\" style=\"display: none\">The first five terms are [latex]\\left\\{1,6, 11, 16, 21\\right\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the first six terms of the sequence.<\/p>\n<p style=\"text-align: center;\">[latex]{a_{n}}=\\begin{cases}2n^{3} & \\text{if }n\\text{ is odd} \\\\[1mm] \\dfrac{5n}{2} & \\text{if }n\\text{ is even}\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q493717\">Show Solution<\/button><\/p>\n<div id=\"q493717\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first six terms are [latex]\\left\\{2,5,54,10,250,15\\right\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding an Explicit Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Pattern Recognition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Look for relationships between terms and their positions<\/li>\n<li class=\"whitespace-normal break-words\">Analyze numerators and denominators separately for fraction sequences<\/li>\n<li class=\"whitespace-normal break-words\">Identify patterns in signs, exponents, or bases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Alternating Sequences:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use [latex](-1)^n[\/latex] or [latex](-1)^{n-1}[\/latex] to represent alternating signs<\/li>\n<li class=\"whitespace-normal break-words\">Don&#8217;t rearrange terms in numerical order<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Structure:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n = [\\text{ expression involving }n][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">May involve algebraic, exponential, or trigonometric functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Always test your formula for the first few terms of the sequence<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\{9;\u221281,729;\u22126,561;59,049\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q806943\">Show Solution<\/button><\/p>\n<div id=\"q806943\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}={\\left(-1\\right)}^{n+1}{9}^{n}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.<\/p>\n<div style=\"text-align: center;\">[latex]\\left\\{-\\dfrac{3}{4},-\\dfrac{9}{8},-\\dfrac{27}{12},-\\dfrac{81}{16},-\\dfrac{243}{20},\\dots\\right\\}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q282988\">Show Solution<\/button><\/p>\n<div id=\"q282988\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=-\\dfrac{{3}^{n}}{4n}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the sequence.<\/p>\n<div style=\"text-align: center;\">[latex]\\left\\{\\dfrac{1}{{e}^{2}}, \\dfrac{1}{e}, 1, e, {e}^{2},...\\right\\}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q804020\">Show Solution<\/button><\/p>\n<div id=\"q804020\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}={e}^{n - 3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence:<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}=\\dfrac{4n}{{\\left(-2\\right)}^{n}}[\/latex]<\/div>\n<div><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q196840\">Show Solution<\/button><\/p>\n<div id=\"q196840\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first five terms are [latex]\\left\\{-2, 2, -\\dfrac{3}{2}, 1,-\\dfrac{5}{8}\\right\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ehhebhaa-f02dpeOaXao\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/f02dpeOaXao?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ehhebhaa-f02dpeOaXao\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780923&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ehhebhaa-f02dpeOaXao&amp;vembed=0&amp;video_id=f02dpeOaXao&amp;video_target=tpm-plugin-ehhebhaa-f02dpeOaXao\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+the+formula+of+alternating+signs+of+a+sequence_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the formula of alternating signs of a sequence\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Sequences Defined by a Recursive Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Recursive Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defines each term using preceding term(s)<\/li>\n<li class=\"whitespace-normal break-words\">Must state initial term(s)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Structure:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Initial condition(s): [latex]a_1 = [\\text{value}][\/latex], ([latex]a_2 = [\\text{value}][\/latex], if needed)<\/li>\n<li class=\"whitespace-normal break-words\">Recursive rule: [latex]a_n = [\\text{expression involving }a_{n-1}, a_{n-2},\\text{ etc.}][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Useful for sequences defined by step-by-step processes<\/li>\n<li class=\"whitespace-normal break-words\">Effective for modeling natural growth and decay phenomena<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparison with Explicit Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Recursive: Defines terms relative to previous terms<\/li>\n<li class=\"whitespace-normal break-words\">Explicit: Defines terms directly based on their position<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">May be computationally intensive for large [latex]n[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Not always easy to find a specific term without calculating all preceding terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the recursive formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{a}_{1}&=2\\\\ {a}_{n}&=2{a}_{n - 1}+1\\text{, for }n\\ge 2\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q378600\">Show Solution<\/button><\/p>\n<div id=\"q378600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebhcdfhg-RjsyEWDEQe0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RjsyEWDEQe0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebhcdfhg-RjsyEWDEQe0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851333&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebhcdfhg-RjsyEWDEQe0&amp;vembed=0&amp;video_id=RjsyEWDEQe0&amp;video_target=tpm-plugin-ebhcdfhg-RjsyEWDEQe0\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Finding+Terms+in+a+Sequence+Given+a+Recursive+Formula_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Finding Terms in a Sequence Given a Recursive Formula\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Factorial Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Factorial:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]n! = n \\times (n-1) \\times (n-2) \\times ... \\times 3 \\times 2 \\times 1[\/latex], for [latex]n \\geq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0![\/latex] is defined as [latex]1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factorials grow very quickly<\/li>\n<li class=\"whitespace-normal break-words\">Often used in combinatorics and probability<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Application in Sequences:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can appear in both explicit and recursive formulas<\/li>\n<li class=\"whitespace-normal break-words\">Often leads to rapidly converging or diverging sequences<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{\\left(n+1\\right)!}{2n}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q953785\">Show Solution<\/button><\/p>\n<div id=\"q953785\" class=\"hidden-answer\" style=\"display: none\">The first five terms are [latex]\\displaystyle \\left\\{1, \\frac{3}{2}, 4,15,72\\right\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Finding the formula of 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