{"id":442,"date":"2019-07-15T22:45:05","date_gmt":"2019-07-15T22:45:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/formulas-for-arithmetic-sequences\/"},"modified":"2019-07-15T22:45:05","modified_gmt":"2019-07-15T22:45:05","slug":"formulas-for-arithmetic-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/formulas-for-arithmetic-sequences\/","title":{"raw":"Formulas for Arithmetic Sequences","rendered":"Formulas for Arithmetic Sequences"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Write an explicit formula for an arithmetic sequence.<\/li>\n \t<li>Write a recursive formula for the&nbsp;arithmetic sequence.<\/li>\n<\/ul>\n<\/div>\n\n<h2>Using Explicit Formulas for Arithmetic Sequences<\/h2>\nWe can think of an <strong>arithmetic sequence<\/strong> as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\nTo find the <em>y<\/em>-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222150\/CNX_Precalc_Figure_11_02_0062.jpg\" alt=\"A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.\">\n\nThe common difference is [latex]-50[\/latex] , so the sequence represents a linear function with a slope of [latex]-50[\/latex] . To find the [latex]y[\/latex] -intercept, we subtract [latex]-50[\/latex] from [latex]200:200-\\left(-50\\right)=200+50=250[\/latex] . You can also find the [latex]y[\/latex] -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222152\/CNX_Precalc_Figure_11_02_0072.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"731\" height=\"250\">\n\nRecall the slope-intercept form of a line is [latex]y=mx+b[\/latex]. When dealing with sequences, we use [latex]{a}_{n}[\/latex] in place of [latex]y[\/latex] and [latex]n[\/latex] in place of [latex]x[\/latex]. If we know the slope and vertical intercept of the function, we can substitute them for [latex]m[\/latex] and [latex]b[\/latex] in the slope-intercept form of a line. Substituting [latex]-50[\/latex] for the slope and [latex]250[\/latex] for the vertical intercept, we get the following equation:\n<p style=\"text-align: center\">[latex]{a}_{n}=-50n+250[\/latex]<\/p>\n\n<div class=\"textbox examples\">\n<h3>sequences as linear functions<\/h3>\nWe've seen several graphs of sequence terms in this module so far. If you think of [latex]n[\/latex] representing the input of the function of an arithmetic sequence and [latex]a_n[\/latex] as the output of the function, it may help you to better visualize the arithmetic sequence as a linear function of the form&nbsp;[latex]y=mx+b[\/latex], or using sequence notation, [latex]a_n=dn+a_0[\/latex] where each point on the graph is of the form [latex]\\left(n, a_n\\right)[\/latex] and the common difference gives us the slope of the line.\n\nNote that if we let [latex]n=0[\/latex] in the explicit form [latex]a_n=a_1+d(n-1)[\/latex], we obtain the statement [latex]a_0=a_1-d[\/latex]. That statement tells us that the vertical intercept [latex]a_0[\/latex] can be found by subtracting the common difference from the first term.\n\n<\/div>\nWe do not need to find the vertical intercept to write an <strong>explicit formula<\/strong> for an arithmetic sequence. Another explicit formula for this sequence is [latex]{a}_{n}=200 - 50\\left(n - 1\\right)[\/latex] , which simplifies to [latex]{a}_{n}=-50n+250[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: Explicit Formula for an Arithmetic Sequence<\/h3>\nAn explicit formula for the [latex]n\\text{th}[\/latex] term of an arithmetic sequence is given by\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/h3>\n<ol>\n \t<li>Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the <em>n<\/em>th Term Explicit Formula for an Arithmetic Sequence<\/h3>\nWrite an explicit formula for the arithmetic sequence.\n\n[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\ldots \\right\\}[\/latex]\n\n[reveal-answer q=\"533579\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"533579\"]\n\nThe common difference can be found by subtracting the first term from the second term.\n<p style=\"text-align: center\">[latex]\\begin{align}d&amp;={a}_{2}-{a}_{1} \\\\ &amp; =12 - 2 \\\\ &amp; =10 \\end{align}[\/latex]<\/p>\nThe common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{n}=2+10\\left(n - 1\\right) \\\\ &amp;{a}_{n}=10n - 8 \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nThe graph of this sequence shows a slope of 10 and a vertical intercept of [latex]-8[\/latex] .\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222154\/CNX_Precalc_Figure_11_02_0082.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"276\">[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite an explicit formula for the following arithmetic sequence.\n[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]\n\n[reveal-answer q=\"524968\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"524968\"]\n\n[latex]{a}_{n}=53 - 3n[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n<\/div>\nSome arithmetic sequences are defined in terms of the previous term using a <strong>recursive formula<\/strong>. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{n}={a}_{n - 1}+d &amp;&amp; n\\ge 2 \\end{align}[\/latex]<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Recursive Formula for an Arithmetic Sequence<\/h3>\nThe recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is:\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{n}={a}_{n - 1}+d &amp;&amp; n\\ge 2 \\end{align}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an arithmetic sequence, write its recursive formula.<\/h3>\n<ol>\n \t<li>Subtract any term from the subsequent term to find the common difference.<\/li>\n \t<li>State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nEither the explicit or the recursive form may be used to describe an arithmetic sequence in which the first term is known. Practice using both of them on the examples in this section.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Recursive Formula for an Arithmetic Sequence<\/h3>\nWrite a <strong>recursive formula<\/strong> for the&nbsp;<strong>arithmetic sequence<\/strong>.\n\n[latex]\\left\\{-18,-7,4,15,26, \\ldots \\right\\}[\/latex]\n\n[reveal-answer q=\"265289\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"265289\"]\n\nThe first term is given as [latex]-18[\/latex] . The common difference can be found by subtracting the first term from the second term.\n<p style=\"text-align: center\">[latex]d=-7-\\left(-18\\right)=11[\/latex]<\/p>\nSubstitute the initial term and the common difference into the recursive formula for arithmetic sequences.\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{1}=-18 \\\\ &amp;{a}_{n}={a}_{n - 1}+11,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nWe see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222148\/CNX_Precalc_Figure_11_02_0052.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"250\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Do we have to subtract the first term from the second term to find the common difference?<\/h3>\n<em> No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite a recursive formula for the arithmetic sequence.\n<p style=\"text-align: center\">[latex]\\left\\{25,37,49,61, \\dots \\right\\}[\/latex]<\/p>\n[reveal-answer q=\"518516\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"518516\"]\n\n[latex]\\begin{align}&amp;{a}_{1}=25 \\\\ &amp;{a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2 \\end{align}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n<\/div>\n<h2>Find the Number of Terms in an Arithmetic Sequence<\/h2>\nExplicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.\n<div class=\"textbox\">\n<h3>How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/h3>\n<ol>\n \t<li>Find the common difference [latex]d[\/latex].<\/li>\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n \t<li>Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nRecall that we only need one equation in one unknown to solve for it. Given the explicit form of an arithmetic sequence, [latex]a_n=a_1+d(n-1)[\/latex], if we can substitute known values in for all but one component, we can solve for the missing one.\n\nIn this case, we are given that a certain finite sequence is arithmetic and we know the first term [latex]a_1[\/latex] and the final term [latex]a_n[\/latex]. We are able to calculate the common difference from any two consecutive terms, which we are given. Substituting these known values into the explicit formula allows us to solve for the unknown value for the number of terms [latex]n[\/latex] without having to generate them.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Terms in a Finite Arithmetic Sequence<\/h3>\nFind the number of terms in the <strong>finite arithmetic sequence<\/strong>.\n[latex]\\left\\{8,1,-6, \\dots ,-41\\right\\}[\/latex]\n\n[reveal-answer q=\"738207\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"738207\"]\n\nThe common difference can be found by subtracting the first term from the second term.\n<p style=\"text-align: center\">[latex]1 - 8=-7[\/latex]<\/p>\nThe common difference is [latex]-7[\/latex] . Substitute the common difference and the initial term of the sequence into the\n\n[latex]n\\text{th}[\/latex] term formula and simplify.\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}+d\\left(n - 1\\right) \\\\ &amp;{a}_{n}=8+-7\\left(n - 1\\right) \\\\ &amp;{a}_{n}=15 - 7n \\end{align}[\/latex]<\/p>\nSubstitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]\n<p style=\"text-align: center\">[latex]\\begin{align}-41&amp;=15 - 7n \\\\ 8&amp;=n \\end{align}[\/latex]<\/p>\nThere are eight terms in the sequence.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the number of terms in the finite arithmetic sequence.\n[latex]\\left\\{6\\text{, }11\\text{, }16\\text{, }...\\text{, }56\\right\\}[\/latex]\n\n[reveal-answer q=\"35032\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"35032\"]\n\nThere are 11 terms in the sequence.\n\n[\/hidden-answer]\n\n[ohm_question]5834[\/ohm_question]\n\n<\/div>\nIn the following video lesson, we present a recap of some of the concepts presented about arithmetic sequences up to this point.\n\nhttps:\/\/youtu.be\/jExpsJTu9o8\n<h2>Solving Application Problems with Arithmetic Sequences<\/h2>\nIn many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:\n[latex]{a}_{n}={a}_{0}+dn[\/latex]\n<div class=\"textbox examples\">\n<h3>Tip for success<\/h3>\nSee the red box SEQUENCES AS LINEAR FUNCTIONS at the start of this section for a derivation and explanation of this formula.\n\nKeep in mind as you work through the example and practice problems that an arithmetic sequence may be represented as a linear function with input [latex]n[\/latex], output [latex]a_n[\/latex], and common difference (or slope) of [latex]d[\/latex].\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Arithmetic Sequences<\/h3>\nA five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.\n<ol>\n \t<li>Write a formula for the child\u2019s weekly allowance in a given year.<\/li>\n \t<li>What will the child\u2019s allowance be when he is 16 years old?<\/li>\n<\/ol>\n[reveal-answer q=\"752686\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"752686\"]\n<ol>\n \t<li>The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:\n[latex]{A}_{n}=1+2n[\/latex]<\/li>\n \t<li>We can find the number of years since age 5 by subtracting.\n[latex]16 - 5=11[\/latex]\nWe are looking for the child\u2019s allowance after 11 years. Substitute 11 into the formula to find the child\u2019s allowance at age 16.\n[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]\nThe child\u2019s allowance at age 16 will be $23 per week.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?\n\n[reveal-answer q=\"356014\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"356014\"]\n\nThe formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29759&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write an explicit formula for an arithmetic sequence.<\/li>\n<li>Write a recursive formula for the&nbsp;arithmetic sequence.<\/li>\n<\/ul>\n<\/div>\n<h2>Using Explicit Formulas for Arithmetic Sequences<\/h2>\n<p>We can think of an <strong>arithmetic sequence<\/strong> as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.<\/p>\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\n<p>To find the <em>y<\/em>-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222150\/CNX_Precalc_Figure_11_02_0062.jpg\" alt=\"A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.\" \/><\/p>\n<p>The common difference is [latex]-50[\/latex] , so the sequence represents a linear function with a slope of [latex]-50[\/latex] . To find the [latex]y[\/latex] -intercept, we subtract [latex]-50[\/latex] from [latex]200:200-\\left(-50\\right)=200+50=250[\/latex] . You can also find the [latex]y[\/latex] -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222152\/CNX_Precalc_Figure_11_02_0072.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"731\" height=\"250\" \/><\/p>\n<p>Recall the slope-intercept form of a line is [latex]y=mx+b[\/latex]. When dealing with sequences, we use [latex]{a}_{n}[\/latex] in place of [latex]y[\/latex] and [latex]n[\/latex] in place of [latex]x[\/latex]. If we know the slope and vertical intercept of the function, we can substitute them for [latex]m[\/latex] and [latex]b[\/latex] in the slope-intercept form of a line. Substituting [latex]-50[\/latex] for the slope and [latex]250[\/latex] for the vertical intercept, we get the following equation:<\/p>\n<p style=\"text-align: center\">[latex]{a}_{n}=-50n+250[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>sequences as linear functions<\/h3>\n<p>We&#8217;ve seen several graphs of sequence terms in this module so far. If you think of [latex]n[\/latex] representing the input of the function of an arithmetic sequence and [latex]a_n[\/latex] as the output of the function, it may help you to better visualize the arithmetic sequence as a linear function of the form&nbsp;[latex]y=mx+b[\/latex], or using sequence notation, [latex]a_n=dn+a_0[\/latex] where each point on the graph is of the form [latex]\\left(n, a_n\\right)[\/latex] and the common difference gives us the slope of the line.<\/p>\n<p>Note that if we let [latex]n=0[\/latex] in the explicit form [latex]a_n=a_1+d(n-1)[\/latex], we obtain the statement [latex]a_0=a_1-d[\/latex]. That statement tells us that the vertical intercept [latex]a_0[\/latex] can be found by subtracting the common difference from the first term.<\/p>\n<\/div>\n<p>We do not need to find the vertical intercept to write an <strong>explicit formula<\/strong> for an arithmetic sequence. Another explicit formula for this sequence is [latex]{a}_{n}=200 - 50\\left(n - 1\\right)[\/latex] , which simplifies to [latex]{a}_{n}=-50n+250[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Explicit Formula for an Arithmetic Sequence<\/h3>\n<p>An explicit formula for the [latex]n\\text{th}[\/latex] term of an arithmetic sequence is given by<\/p>\n<p style=\"text-align: center\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/h3>\n<ol>\n<li>Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\n<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the <em>n<\/em>th Term Explicit Formula for an Arithmetic Sequence<\/h3>\n<p>Write an explicit formula for the arithmetic sequence.<\/p>\n<p>[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\ldots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q533579\">Show Solution<\/span><\/p>\n<div id=\"q533579\" class=\"hidden-answer\" style=\"display: none\">\n<p>The common difference can be found by subtracting the first term from the second term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}d&={a}_{2}-{a}_{1} \\\\ & =12 - 2 \\\\ & =10 \\end{align}[\/latex]<\/p>\n<p>The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{n}=2+10\\left(n - 1\\right) \\\\ &{a}_{n}=10n - 8 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of this sequence shows a slope of 10 and a vertical intercept of [latex]-8[\/latex] .<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222154\/CNX_Precalc_Figure_11_02_0082.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"276\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write an explicit formula for the following arithmetic sequence.<br \/>\n[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q524968\">Show Solution<\/span><\/p>\n<div id=\"q524968\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=53 - 3n[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm23521\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&#38;theme=oea&#38;iframe_resize_id=ohm23521&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Some arithmetic sequences are defined in terms of the previous term using a <strong>recursive formula<\/strong>. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{n}={a}_{n - 1}+d && n\\ge 2 \\end{align}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Recursive Formula for an Arithmetic Sequence<\/h3>\n<p>The recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{n}={a}_{n - 1}+d && n\\ge 2 \\end{align}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an arithmetic sequence, write its recursive formula.<\/h3>\n<ol>\n<li>Subtract any term from the subsequent term to find the common difference.<\/li>\n<li>State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Either the explicit or the recursive form may be used to describe an arithmetic sequence in which the first term is known. Practice using both of them on the examples in this section.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Recursive Formula for an Arithmetic Sequence<\/h3>\n<p>Write a <strong>recursive formula<\/strong> for the&nbsp;<strong>arithmetic sequence<\/strong>.<\/p>\n<p>[latex]\\left\\{-18,-7,4,15,26, \\ldots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265289\">Show Solution<\/span><\/p>\n<div id=\"q265289\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is given as [latex]-18[\/latex] . The common difference can be found by subtracting the first term from the second term.<\/p>\n<p style=\"text-align: center\">[latex]d=-7-\\left(-18\\right)=11[\/latex]<\/p>\n<p>Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{1}=-18 \\\\ &{a}_{n}={a}_{n - 1}+11,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222148\/CNX_Precalc_Figure_11_02_0052.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"250\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Do we have to subtract the first term from the second term to find the common difference?<\/h3>\n<p><em> No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q518516\">Show Solution<\/span><\/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<p><iframe loading=\"lazy\" id=\"ohm23521\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&#38;theme=oea&#38;iframe_resize_id=ohm23521&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Find the Number of Terms in an Arithmetic Sequence<\/h2>\n<p>Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/h3>\n<ol>\n<li>Find the common difference [latex]d[\/latex].<\/li>\n<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<li>Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Recall that we only need one equation in one unknown to solve for it. Given the explicit form of an arithmetic sequence, [latex]a_n=a_1+d(n-1)[\/latex], if we can substitute known values in for all but one component, we can solve for the missing one.<\/p>\n<p>In this case, we are given that a certain finite sequence is arithmetic and we know the first term [latex]a_1[\/latex] and the final term [latex]a_n[\/latex]. We are able to calculate the common difference from any two consecutive terms, which we are given. Substituting these known values into the explicit formula allows us to solve for the unknown value for the number of terms [latex]n[\/latex] without having to generate them.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Number of Terms in a Finite Arithmetic Sequence<\/h3>\n<p>Find the number of terms in the <strong>finite arithmetic sequence<\/strong>.<br \/>\n[latex]\\left\\{8,1,-6, \\dots ,-41\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738207\">Show Solution<\/span><\/p>\n<div id=\"q738207\" class=\"hidden-answer\" style=\"display: none\">\n<p>The common difference can be found by subtracting the first term from the second term.<\/p>\n<p style=\"text-align: center\">[latex]1 - 8=-7[\/latex]<\/p>\n<p>The common difference is [latex]-7[\/latex] . Substitute the common difference and the initial term of the sequence into the<\/p>\n<p>[latex]n\\text{th}[\/latex] term formula and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{a}_{n}={a}_{1}+d\\left(n - 1\\right) \\\\ &{a}_{n}=8+-7\\left(n - 1\\right) \\\\ &{a}_{n}=15 - 7n \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}-41&=15 - 7n \\\\ 8&=n \\end{align}[\/latex]<\/p>\n<p>There are eight terms in the sequence.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35032\">Show Solution<\/span><\/p>\n<div id=\"q35032\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 11 terms in the sequence.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm5834\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5834&theme=oea&iframe_resize_id=ohm5834&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Arithmetic Sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jExpsJTu9o8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Solving Application Problems with Arithmetic Sequences<\/h2>\n<p>In many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:<br \/>\n[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Tip for success<\/h3>\n<p>See the red box SEQUENCES AS LINEAR FUNCTIONS at the start of this section for a derivation and explanation of this formula.<\/p>\n<p>Keep in mind as you work through the example and practice problems that an arithmetic sequence may be represented as a linear function with input [latex]n[\/latex], output [latex]a_n[\/latex], and common difference (or slope) of [latex]d[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Arithmetic Sequences<\/h3>\n<p>A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.<\/p>\n<ol>\n<li>Write a formula for the child\u2019s weekly allowance in a given year.<\/li>\n<li>What will the child\u2019s allowance be when he is 16 years old?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q752686\">Show Solution<\/span><\/p>\n<div id=\"q752686\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:<br \/>\n[latex]{A}_{n}=1+2n[\/latex]<\/li>\n<li>We can find the number of years since age 5 by subtracting.<br \/>\n[latex]16 - 5=11[\/latex]<br \/>\nWe are looking for the child\u2019s allowance after 11 years. Substitute 11 into the formula to find the child\u2019s allowance at age 16.<br \/>\n[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<br \/>\nThe child\u2019s allowance at age 16 will be $23 per week.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q356014\">Show Solution<\/span><\/p>\n<div id=\"q356014\" class=\"hidden-answer\" style=\"display: none\">\n<p>The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm29759\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29759&#38;theme=oea&#38;iframe_resize_id=ohm29759&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-442\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 29759. <strong>Authored by<\/strong>: McClure,Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 23521. <strong>Authored by<\/strong>: Shahbazian,Roy, mb McClure,Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Arithmetic Sequences . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jExpsJTu9o8\">https:\/\/youtu.be\/jExpsJTu9o8<\/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":17533,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 29759\",\"author\":\"McClure,Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 23521\",\"author\":\"Shahbazian,Roy, mb McClure,Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Arithmetic Sequences \",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/jExpsJTu9o8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"18476a5f-8c61-4ac7-aff0-4203ac0cf42e","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-442","chapter","type-chapter","status-publish","hentry"],"part":431,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/442","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/442\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/431"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/442\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=442"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=442"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=442"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}