{"id":1484,"date":"2025-07-25T02:09:37","date_gmt":"2025-07-25T02:09:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1484"},"modified":"2026-03-24T07:13:31","modified_gmt":"2026-03-24T07:13:31","slug":"arithmetic-sequences-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-fresh-take\/","title":{"raw":"Arithmetic Sequences: Fresh Take","rendered":"Arithmetic Sequences: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the common difference for an arithmetic sequence.<\/li>\r\n \t<li>Write the formula for an arithmetic sequence.<\/li>\r\n \t<li>Use arithmetic sequences to solve realistic scenarios<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Terms of an Arithmetic Sequence<\/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:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A sequence where the difference between consecutive terms is constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This constant difference is called the common difference, denoted '[latex]d[\/latex]'<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form: [latex]{a_n} = {a_1, a_1+d, a_1+2d, a_1+3d, ...}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear growth<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphed as points on a straight line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Real-world Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Depreciation (straight-line method)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Linear salary increases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Arithmetic interest<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Is the given sequence arithmetic? If so, find the common difference.\r\n<p style=\"text-align: center;\">[latex]\\left\\{18,16,14,12,10,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"463836\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"463836\"]\r\n\r\nThe sequence is arithmetic. The common difference is [latex]-2[\/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-edhahfab-uACt9OntiLo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uACt9OntiLo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-edhahfab-uACt9OntiLo\" 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=12851334&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-edhahfab-uACt9OntiLo&amp;vembed=0&amp;video_id=uACt9OntiLo&amp;video_target=tpm-plugin-edhahfab-uACt9OntiLo\" 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\/Determine+a+Terms+Position+in+an+Arithmetic+Sequence_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine a Terms Position in an Arithmetic Sequence\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Terms of Arithmetic Sequences<\/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\">General Term Formula: [latex]a_n = a_1 + (n - 1)d[\/latex] Where:\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[\/latex] is the nth term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\r\n<\/ul>\r\n<\/li>\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\">The coefficient of [latex]d[\/latex] is always one less than the term number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reverse Engineering:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Given any two terms, we can find the common difference and first term<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">List the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .[reveal-answer q=\"880961\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880961\"][latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Using Explicit Formulas for Arithmetic Sequences<\/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\">Explicit Formula: [latex]a_n = a_1 + d(n - 1)[\/latex] Where:\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[\/latex] is the nth term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Deriving the Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Observe the pattern: [latex]a_1, a_1 + d, a_1 + 2d, a_1 + 3d, ...[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Generalize to [latex]a_1 + (n-1)d[\/latex] for the nth term<\/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\">Quickly find any term without calculating all previous terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyze long-term behavior of sequences<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Connection to Linear Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The explicit formula represents a linear function when n is treated as a continuous variable<\/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 following arithmetic sequence.\r\n[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex][reveal-answer q=\"524968\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"524968\"][latex]{a}_{n}=53 - 3n[\/latex][\/hidden-answer]<\/section>\r\n<h2>Using Recursive Formulas for Arithmetic Sequences<\/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\">Recursive Formula: [latex]a_n = a_{n-1} + d, \\text{ for } n \\geq 2[\/latex] Where:\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[\/latex] is the nth term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_{n-1}[\/latex] is the previous term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Initial condition: [latex]a_1[\/latex] (first term) must be specified<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\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 based on the previous term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Requires knowing the first term and common difference<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Advantages:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Often mirrors the natural way sequences are generated<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be easier to understand for some types of sequences<\/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\">Can be computationally intensive for finding terms far in the sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May not provide immediate insight into the long-term behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write a recursive formula for the arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{25,37,49,61, \\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"518516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518516\"]\r\n\r\n[latex]\\begin{align}&amp;{a}_{1}=25 \\\\ &amp;{a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2 \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Find the Number of Terms in an Arithmetic Sequence<\/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\">Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">In a finite arithmetic sequence, we can determine the total number of terms if we know the first term, last term, and common difference.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Formula: [latex]a_n = a_1 + d(n - 1)[\/latex] Where:\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[\/latex] is the last term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the number of terms (what we're solving for)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find the common difference<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set up the equation using the first and last terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The solution [latex]n[\/latex] must be a positive integer<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]n[\/latex] is not a whole number, round up to the next integer for the minimum number of terms needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the number of terms in the finite arithmetic sequence.\r\n[latex]\\left\\{6\\text{, }11\\text{, }16\\text{, }...\\text{, }56\\right\\}[\/latex][reveal-answer q=\"35032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35032\"]There are [latex]11[\/latex] terms in the sequence.[\/hidden-answer]<\/section>In the following video lesson, we present a recap of some of the concepts presented about arithmetic sequences up to this point.\r\n\r\n<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-fffefhed-jExpsJTu9o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jExpsJTu9o8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fffefhed-jExpsJTu9o8\" 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=12851335&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fffefhed-jExpsJTu9o8&amp;vembed=0&amp;video_id=jExpsJTu9o8&amp;video_target=tpm-plugin-fffefhed-jExpsJTu9o8\" 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\/Arithmetic+Sequences_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cArithmetic Sequences\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the common difference for an arithmetic sequence.<\/li>\n<li>Write the formula for an arithmetic sequence.<\/li>\n<li>Use arithmetic sequences to solve realistic scenarios<\/li>\n<\/ul>\n<\/section>\n<h2>Terms of an Arithmetic Sequence<\/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:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A sequence where the difference between consecutive terms is constant<\/li>\n<li class=\"whitespace-normal break-words\">This constant difference is called the common difference, denoted &#8216;[latex]d[\/latex]&#8216;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form: [latex]{a_n} = {a_1, a_1+d, a_1+2d, a_1+3d, ...}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear growth<\/li>\n<li class=\"whitespace-normal break-words\">Graphed as points on a straight line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Real-world Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Depreciation (straight-line method)<\/li>\n<li class=\"whitespace-normal break-words\">Linear salary increases<\/li>\n<li class=\"whitespace-normal break-words\">Arithmetic interest<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Is the given sequence arithmetic? If so, find the common difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{18,16,14,12,10,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q463836\">Show Solution<\/button><\/p>\n<div id=\"q463836\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is arithmetic. The common difference is [latex]-2[\/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-edhahfab-uACt9OntiLo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uACt9OntiLo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-edhahfab-uACt9OntiLo\" 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=12851334&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-edhahfab-uACt9OntiLo&amp;vembed=0&amp;video_id=uACt9OntiLo&amp;video_target=tpm-plugin-edhahfab-uACt9OntiLo\" 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\/Determine+a+Terms+Position+in+an+Arithmetic+Sequence_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine a Terms Position in an Arithmetic Sequence\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Terms of Arithmetic Sequences<\/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\">General Term Formula: [latex]a_n = a_1 + (n - 1)d[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the nth term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\n<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\n<\/ul>\n<\/li>\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\">The coefficient of [latex]d[\/latex] is always one less than the term number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reverse Engineering:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Given any two terms, we can find the common difference and first term<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">List the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880961\">Show Solution<\/button><\/p>\n<div id=\"q880961\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Using Explicit Formulas for Arithmetic Sequences<\/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\">Explicit Formula: [latex]a_n = a_1 + d(n - 1)[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the nth term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Deriving the Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Observe the pattern: [latex]a_1, a_1 + d, a_1 + 2d, a_1 + 3d, ...[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Generalize to [latex]a_1 + (n-1)d[\/latex] for the nth term<\/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\">Quickly find any term without calculating all previous terms<\/li>\n<li class=\"whitespace-normal break-words\">Analyze long-term behavior of sequences<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Connection to Linear Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The explicit formula represents a linear function when n is treated as a continuous variable<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the following arithmetic sequence.<br \/>\n[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q524968\">Show Solution<\/button><\/p>\n<div id=\"q524968\" class=\"hidden-answer\" style=\"display: none\">[latex]{a}_{n}=53 - 3n[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Using Recursive Formulas for Arithmetic Sequences<\/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\">Recursive Formula: [latex]a_n = a_{n-1} + d, \\text{ for } n \\geq 2[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the nth term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_{n-1}[\/latex] is the previous term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\n<li class=\"whitespace-normal break-words\">Initial condition: [latex]a_1[\/latex] (first term) must be specified<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defines each term based on the previous term<\/li>\n<li class=\"whitespace-normal break-words\">Requires knowing the first term and common difference<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often mirrors the natural way sequences are generated<\/li>\n<li class=\"whitespace-normal break-words\">Can be easier to understand for some types of sequences<\/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\">Can be computationally intensive for finding terms far in the sequence<\/li>\n<li class=\"whitespace-normal break-words\">May not provide immediate insight into the long-term behavior<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write a recursive formula for the arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{25,37,49,61, \\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q518516\">Show Solution<\/button><\/p>\n<div id=\"q518516\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}&{a}_{1}=25 \\\\ &{a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Find the Number of Terms in an Arithmetic Sequence<\/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\">Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">In a finite arithmetic sequence, we can determine the total number of terms if we know the first term, last term, and common difference.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formula: [latex]a_n = a_1 + d(n - 1)[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the last term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]d[\/latex] is the common difference<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the number of terms (what we&#8217;re solving for)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find the common difference<\/li>\n<li class=\"whitespace-normal break-words\">Set up the equation using the first and last terms<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The solution [latex]n[\/latex] must be a positive integer<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]n[\/latex] is not a whole number, round up to the next integer for the minimum number of terms needed<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the number of terms in the finite arithmetic sequence.<br \/>\n[latex]\\left\\{6\\text{, }11\\text{, }16\\text{, }...\\text{, }56\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q35032\">Show Solution<\/button><\/p>\n<div id=\"q35032\" class=\"hidden-answer\" style=\"display: none\">There are [latex]11[\/latex] terms in the sequence.<\/div>\n<\/div>\n<\/section>\n<p>In the following video lesson, we present a recap of some of the concepts presented about arithmetic sequences up to this point.<\/p>\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-fffefhed-jExpsJTu9o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jExpsJTu9o8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fffefhed-jExpsJTu9o8\" 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=12851335&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fffefhed-jExpsJTu9o8&amp;vembed=0&amp;video_id=jExpsJTu9o8&amp;video_target=tpm-plugin-fffefhed-jExpsJTu9o8\" 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\/Arithmetic+Sequences_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cArithmetic Sequences\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determine a Terms 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