{"id":1859,"date":"2023-10-12T00:32:21","date_gmt":"2023-10-12T00:32:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-arithmetic-sequences\/"},"modified":"2023-10-12T00:32:21","modified_gmt":"2023-10-12T00:32:21","slug":"introduction-arithmetic-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-arithmetic-sequences\/","title":{"raw":"Arithmetic Sequences &amp; Series","rendered":"Arithmetic Sequences &amp; Series"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n \t<li class=\"li2\"><span class=\"s1\">Find the common difference for an arithmetic sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Write terms of an arithmetic sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use a recursive formula for an arithmetic sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use an explicit formula for an arithmetic sequence.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use summation notation.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of an arithmetic series.<\/span><\/li>\n<\/ul>\n<\/div>\nCompanies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.\n\nAs an example consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck\u2019s value.\n<h2>Terms of an Arithmetic Sequence<\/h2>\nThe values of the truck in the example form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence the common difference is \u20133,400.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222135\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\">\n\nThe sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find the subsequent term.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222137\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\">\n<div class=\"textbox\">\n<h3>A General Note: Arithmetic Sequence<\/h3>\nAn <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:\n[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Common Differences<\/h3>\nIs each sequence arithmetic? If so, find the common difference.\n<ol>\n \t<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\n \t<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"717238\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"717238\"]\n\nSubtract each term from the subsequent term to determine whether a common difference exists.\n<ol>\n \t<li>The sequence is not arithmetic because there is no common difference.\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;2-1=1 &amp;&amp; 4-2=2 &amp;&amp; 8-4=4 &amp;&amp; 16-8=8 \\end{align}[\/latex]<\/div><\/li>\n \t<li>The sequence is arithmetic because there is a common difference. The common difference is 4.\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;1-(-3)=4 &amp;&amp; 5-1=4 &amp;&amp; 9-5=4 &amp;&amp; 13-9=4 \\end{align}[\/latex]<\/div><\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\nThe graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, [latex]a[\/latex] is not linear whereas [latex]b[\/latex] is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222143\/CNX_Precalc_Figure_11_02_0032.jpg\" alt=\"Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly.\" width=\"975\" height=\"304\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?<\/h4>\n<em> No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nIs the given sequence arithmetic? If so, find the common difference.\n<p style=\"text-align: center;\">[latex]\\left\\{18,16,14,12,10,\\dots \\right\\}[\/latex]<\/p>\n[reveal-answer q=\"463836\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"463836\"]\n\nThe sequence is arithmetic. The common difference is [latex]-2[\/latex].\n\n[\/hidden-answer]\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nIs the given sequence arithmetic? If so, find the common difference.\n<p style=\"text-align: center;\">[latex]\\left\\{1,3,6,10,15,\\dots \\right\\}[\/latex]<\/p>\n[reveal-answer q=\"598102\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"598102\"]\n\nThe sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<h3>Writing Terms of Arithmetic Sequences<\/h3>\nNow that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/p>\n\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\n<ol>\n \t<li>Add the common difference to the first term to find the second term.<\/li>\n \t<li>Add the common difference to the second term to find the third term.<\/li>\n \t<li>Continue until all of the desired terms are identified.<\/li>\n \t<li>Write the terms separated by commas within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\nWrite the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].\n\n[reveal-answer q=\"654025\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"654025\"]\n\nAdding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.\n\nThe first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]\n<h4>Analysis of the Solution<\/h4>\nAs expected, the graph of the sequence consists of points on a line.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03222146\/CNX_Precalc_Figure_11_02_0042.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"487\" height=\"250\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nList the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .\n\n[reveal-answer q=\"880961\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"880961\"]\n\n[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5832&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given any the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\n<ol>\n \t<li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li>\n \t<li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\nGiven [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex] .\n\n[reveal-answer q=\"644479\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"644479\"]\n\nThe sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex] .\n<p style=\"text-align: center;\">[latex]\\left\\{8,8+d,8+2d,8+3d\\right\\}[\/latex]<\/p>\nWe know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex] .\n\nWe can find the common difference [latex]d[\/latex] .\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &amp;{a}_{4}={a}_{1}+3d \\\\ &amp;{a}_{4}=8+3d &amp;&amp; \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &amp;14=8+3d &amp;&amp; \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &amp;d=2 &amp;&amp; \\text{Solve for the common difference}. \\end{align}[\/latex]<\/p>\nFind the fifth term by adding the common difference to the fourth term.\n<p style=\"text-align: center;\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nNotice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nGiven [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .\n\n[reveal-answer q=\"20007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"20007\"]\n\n[latex]{a}_{2}=2[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5847&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<h2>Formulas for Arithmetic Sequences<\/h2>\n<h3>Using Explicit Formulas for Arithmetic Sequences<\/h3>\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>\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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\n\n<\/div>\n<h3>Find the Number of Terms in an Arithmetic Sequence<\/h3>\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 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<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5834&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"300\"><\/iframe>\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 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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29759&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h3>Arithmetic Series<\/h3>\nJust as we studied special types of sequences, we will look at special types of series. Recall that an <strong>arithmetic sequence<\/strong> is a sequence in which the difference between any two consecutive terms is the <strong>common difference<\/strong>, [latex]d[\/latex]. The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}[\/latex].<\/p>\nWe can also reverse the order of the terms and write the sum as\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}[\/latex].<\/p>\nIf we add these two expressions for the sum of the first [latex]n[\/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&amp;={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n} \\\\ +{S}_{n}&amp;={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1} \\\\ \\hline 2{S}_{n}&amp;=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right) \\end{align}[\/latex]<\/p>\nBecause there are [latex]n[\/latex] terms in the series, we can simplify this sum to\n<p style=\"text-align: center;\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right)[\/latex].<\/p>\nWe divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n<p style=\"text-align: left;\">This is generally referred to as the <strong>Partial Sum<\/strong> of the series.<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Partial Sum of&nbsp;an Arithmetic Series<\/h3>\nAn <strong>arithmetic series<\/strong> is the sum of the terms of an arithmetic sequence. The formula for the partial sum of an arithmetic sequence is\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given terms of an arithmetic series, find the partial sum<\/h3>\n<ol>\n \t<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}[\/latex].<\/li>\n \t<li>Determine [latex]n[\/latex].<\/li>\n \t<li>Substitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex].<\/li>\n \t<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the partial sum&nbsp;of an Arithmetic Series<\/h3>\nFind the partial sum of each arithmetic series.\n<ol>\n \t<li>[latex]5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32[\/latex]<\/li>\n \t<li>[latex]20 + 15 + 10 + \\dots + -50[\/latex]<\/li>\n \t<li>[latex]\\sum\\limits _{k=1}^{12}3k - 8[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"470866\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"470866\"]\n<ol>\n \t<li>[latex]\\begin{align} \\\\ &amp;{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &amp;{S}_{10}=\\dfrac{10\\left(5+32\\right)}{2}=185 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n \t<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].\n[latex]\\begin{align}\\\\ {a}_{n}&amp;={a}_{1}+\\left(n - 1\\right)d \\\\ -50&amp;=20+\\left(n - 1\\right)\\left(-5\\right) \\\\ -70&amp;=\\left(n - 1\\right)\\left(-5\\right) \\\\ 14&amp;=n - 1 \\\\ 15&amp;=n \\\\ \\text{ }\\end{align}[\/latex]\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.\n[latex]\\begin{align}\\\\ &amp;{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &amp;{S}_{15}=\\dfrac{15\\left(20 - 50\\right)}{2}=-225 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n \t<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.\n[latex]\\begin{align}\\\\ {a}_{k}&amp;=3k - 8 \\\\ {a}_{1}&amp;=3\\left(1\\right)-8=-5 \\\\ \\text{ }\\end{align}[\/latex]\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.\n[latex]\\begin{align} \\\\{a}_{k}&amp;=3k - 8 \\\\ {a}_{12}&amp;=3\\left(12\\right)-8=28 \\\\ \\text{ }\\end{align}[\/latex]\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.\n[latex]\\begin{align}\\\\{S}_{n}&amp;=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{12}&amp;=\\dfrac{12\\left(-5+28\\right)}{2}=138 \\end{align}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nUse the formula to find the partial sum of each arithmetic series.\n\n[latex]1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4[\/latex]\n\n[reveal-answer q=\"649728\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"649728\"]\n\n[latex]26.4[\/latex][\/hidden-answer]\n\n[latex]12+21+29\\dots + 69[\/latex]\n\n[reveal-answer q=\"617640\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"617640\"]\n\n[latex]328[\/latex]\n\n[\/hidden-answer]\n\n[latex]\\sum\\limits _{k=1}^{10}5 - 6k[\/latex]\n\n[reveal-answer q=\"794771\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"794771\"]\n\n[latex]-280[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Arithmetic Series<\/h3>\nOn the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?\n\n[reveal-answer q=\"455757\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"455757\"]\n\nThis problem can be modeled by an arithmetic series with [latex]{a}_{1}=\\frac{1}{2}[\/latex] and [latex]d=\\frac{1}{4}[\/latex]. We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8[\/latex], and we are looking for [latex]{S}_{8}[\/latex]. To find [latex]{a}_{8}[\/latex], we can use the explicit formula for an arithmetic sequence.\n<p style=\"text-align: center;\">[latex]\\begin{align} {a}_{n}&amp;={a}_{1}+d\\left(n - 1\\right) \\\\ {a}_{8}&amp;=\\dfrac{1}{2}+\\dfrac{1}{4}\\left(8 - 1\\right)=\\dfrac{9}{4} \\end{array}[\/latex]<\/p>\nWe can now use the formula for arithmetic series.\n<p style=\"text-align: center;\">[latex]\\begin{align} {S}_{n}&amp;=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{8}&amp;=\\dfrac{8\\left(\\frac{1}{2}+\\dfrac{9}{4}\\right)}{2}=11 \\end{align}[\/latex]<\/p>\nShe will have walked a total of 11 miles.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?\n\n[reveal-answer q=\"454197\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"454197\"]\n\n$2,025\n\n[\/hidden-answer]\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5867&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\n\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>recursive formula for nth term of an arithmetic sequence<\/td>\n<td>[latex]{a}_{n}={a}_{n - 1}+d \\text{ for } n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for nth term of an arithmetic sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of the first [latex]n[\/latex]\nterms of an arithmetic series<\/td>\n<td>[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.<\/li>\n \t<li>The constant between two consecutive terms is called the common difference.<\/li>\n \t<li>The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.<\/li>\n \t<li>The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.<\/li>\n \t<li>A recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\\ge 2[\/latex].<\/li>\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n \t<li>An explicit formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n \t<li>An explicit formula can be used to find the number of terms in a sequence.<\/li>\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[\/latex].<\/li>\n \t<li>The sum of the terms in a sequence is called a series.<\/li>\n \t<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.<\/li>\n \t<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\n \t<li>The sum of the first [latex]n[\/latex] terms of an arithmetic series can be found using a formula.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>arithmetic sequence<\/strong> a sequence in which the difference between any two consecutive terms is a constant\n\n<strong>common difference<\/strong> the difference between any two consecutive terms in an arithmetic sequence\n\n<strong>arithmetic series<\/strong> the sum of the terms in an arithmetic sequence\n\n<strong>nth partial sum<\/strong> the sum of the first [latex]n[\/latex] terms of a sequence\n\n<strong>series<\/strong> the sum of the terms in a sequence\n\n<strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Find the common difference for an arithmetic sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Write terms of an arithmetic sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use a recursive formula for an arithmetic sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use an explicit formula for an arithmetic sequence.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use summation notation.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the formula for the sum of the \ufb01rst <\/span><span class=\"s4\"><i>n&nbsp;<\/i><\/span><span class=\"s1\">terms of an arithmetic series.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.<\/p>\n<p>As an example consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck\u2019s value.<\/p>\n<h2>Terms of an Arithmetic Sequence<\/h2>\n<p>The values of the truck in the example form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence the common difference is \u20133,400.<\/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\/03222135\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" \/><\/p>\n<p>The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find the subsequent term.<\/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\/03222137\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Arithmetic Sequence<\/h3>\n<p>An <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:<br \/>\n[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Common Differences<\/h3>\n<p>Is each sequence arithmetic? If so, find the common difference.<\/p>\n<ol>\n<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\n<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717238\">Show Solution<\/span><\/p>\n<div id=\"q717238\" class=\"hidden-answer\" style=\"display: none\">\n<p>Subtract each term from the subsequent term to determine whether a common difference exists.<\/p>\n<ol>\n<li>The sequence is not arithmetic because there is no common difference.\n<div style=\"text-align: center;\">[latex]\\begin{align}&2-1=1 && 4-2=2 && 8-4=4 && 16-8=8 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>The sequence is arithmetic because there is a common difference. The common difference is 4.\n<div style=\"text-align: center;\">[latex]\\begin{align}&1-(-3)=4 && 5-1=4 && 9-5=4 && 13-9=4 \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, [latex]a[\/latex] is not linear whereas [latex]b[\/latex] is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.<\/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\/03222143\/CNX_Precalc_Figure_11_02_0032.jpg\" alt=\"Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly.\" width=\"975\" height=\"304\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?<\/h4>\n<p><em> No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q463836\">Show Solution<\/span><\/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<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the given sequence arithmetic? If so, find the common difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{1,3,6,10,15,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q598102\">Show Solution<\/span><\/p>\n<div id=\"q598102\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Writing Terms of Arithmetic Sequences<\/h3>\n<p>Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\n<ol>\n<li>Add the common difference to the first term to find the second term.<\/li>\n<li>Add the common difference to the second term to find the third term.<\/li>\n<li>Continue until all of the desired terms are identified.<\/li>\n<li>Write the terms separated by commas within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Write the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654025\">Show Solution<\/span><\/p>\n<div id=\"q654025\" class=\"hidden-answer\" style=\"display: none\">\n<p>Adding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.<\/p>\n<p>The first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>As expected, the graph of the sequence consists of points on a line.<\/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\/03222146\/CNX_Precalc_Figure_11_02_0042.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"487\" height=\"250\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880961\">Show Solution<\/span><\/p>\n<div id=\"q880961\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5832&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given any the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\n<ol>\n<li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li>\n<li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Given [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q644479\">Show Solution<\/span><\/p>\n<div id=\"q644479\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex] .<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{8,8+d,8+2d,8+3d\\right\\}[\/latex]<\/p>\n<p>We know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex] .<\/p>\n<p>We can find the common difference [latex]d[\/latex] .<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &{a}_{4}={a}_{1}+3d \\\\ &{a}_{4}=8+3d && \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &14=8+3d && \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &d=2 && \\text{Solve for the common difference}. \\end{align}[\/latex]<\/p>\n<p>Find the fifth term by adding the common difference to the fourth term.<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q20007\">Show Solution<\/span><\/p>\n<div id=\"q20007\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5847&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Formulas for Arithmetic Sequences<\/h2>\n<h3>Using Explicit Formulas for Arithmetic Sequences<\/h3>\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<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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23521&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h3>Find the Number of Terms in an Arithmetic Sequence<\/h3>\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 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=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5834&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"300\"><\/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 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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29759&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Arithmetic Series<\/h3>\n<p>Just as we studied special types of sequences, we will look at special types of series. Recall that an <strong>arithmetic sequence<\/strong> is a sequence in which the difference between any two consecutive terms is the <strong>common difference<\/strong>, [latex]d[\/latex]. The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}[\/latex].<\/p>\n<p>We can also reverse the order of the terms and write the sum as<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}[\/latex].<\/p>\n<p>If we add these two expressions for the sum of the first [latex]n[\/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{S}_{n}&={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n} \\\\ +{S}_{n}&={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1} \\\\ \\hline 2{S}_{n}&=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right) \\end{align}[\/latex]<\/p>\n<p>Because there are [latex]n[\/latex] terms in the series, we can simplify this sum to<\/p>\n<p style=\"text-align: center;\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right)[\/latex].<\/p>\n<p>We divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n<p style=\"text-align: left;\">This is generally referred to as the <strong>Partial Sum<\/strong> of the series.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Partial Sum of&nbsp;an Arithmetic Series<\/h3>\n<p>An <strong>arithmetic series<\/strong> is the sum of the terms of an arithmetic sequence. The formula for the partial sum of an arithmetic sequence is<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given terms of an arithmetic series, find the partial sum<\/h3>\n<ol>\n<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}[\/latex].<\/li>\n<li>Determine [latex]n[\/latex].<\/li>\n<li>Substitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex].<\/li>\n<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the partial sum&nbsp;of an Arithmetic Series<\/h3>\n<p>Find the partial sum of each arithmetic series.<\/p>\n<ol>\n<li>[latex]5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32[\/latex]<\/li>\n<li>[latex]20 + 15 + 10 + \\dots + -50[\/latex]<\/li>\n<li>[latex]\\sum\\limits _{k=1}^{12}3k - 8[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q470866\">Show Solution<\/span><\/p>\n<div id=\"q470866\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{align} \\\\ &{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &{S}_{10}=\\dfrac{10\\left(5+32\\right)}{2}=185 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].<br \/>\n[latex]\\begin{align}\\\\ {a}_{n}&={a}_{1}+\\left(n - 1\\right)d \\\\ -50&=20+\\left(n - 1\\right)\\left(-5\\right) \\\\ -70&=\\left(n - 1\\right)\\left(-5\\right) \\\\ 14&=n - 1 \\\\ 15&=n \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align}\\\\ &{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ &{S}_{15}=\\dfrac{15\\left(20 - 50\\right)}{2}=-225 \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{align}\\\\ {a}_{k}&=3k - 8 \\\\ {a}_{1}&=3\\left(1\\right)-8=-5 \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{align} \\\\{a}_{k}&=3k - 8 \\\\ {a}_{12}&=3\\left(12\\right)-8=28 \\\\ \\text{ }\\end{align}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{align}\\\\{S}_{n}&=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{12}&=\\dfrac{12\\left(-5+28\\right)}{2}=138 \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the formula to find the partial sum of each arithmetic series.<\/p>\n<p>[latex]1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q649728\">Show Solution<\/span><\/p>\n<div id=\"q649728\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]26.4[\/latex]<\/p><\/div>\n<\/div>\n<p>[latex]12+21+29\\dots + 69[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617640\">Show Solution<\/span><\/p>\n<div id=\"q617640\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]328[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>[latex]\\sum\\limits _{k=1}^{10}5 - 6k[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q794771\">Show Solution<\/span><\/p>\n<div id=\"q794771\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-280[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Arithmetic Series<\/h3>\n<p>On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455757\">Show Solution<\/span><\/p>\n<div id=\"q455757\" class=\"hidden-answer\" style=\"display: none\">\n<p>This problem can be modeled by an arithmetic series with [latex]{a}_{1}=\\frac{1}{2}[\/latex] and [latex]d=\\frac{1}{4}[\/latex]. We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8[\/latex], and we are looking for [latex]{S}_{8}[\/latex]. To find [latex]{a}_{8}[\/latex], we can use the explicit formula for an arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} {a}_{n}&={a}_{1}+d\\left(n - 1\\right) \\\\ {a}_{8}&=\\dfrac{1}{2}+\\dfrac{1}{4}\\left(8 - 1\\right)=\\dfrac{9}{4} \\end{array}[\/latex]<\/p>\n<p>We can now use the formula for arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} {S}_{n}&=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2} \\\\ {S}_{8}&=\\dfrac{8\\left(\\frac{1}{2}+\\dfrac{9}{4}\\right)}{2}=11 \\end{align}[\/latex]<\/p>\n<p>She will have walked a total of 11 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q454197\">Show Solution<\/span><\/p>\n<div id=\"q454197\" class=\"hidden-answer\" style=\"display: none\">\n<p>$2,025<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5867&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>recursive formula for nth term of an arithmetic sequence<\/td>\n<td>[latex]{a}_{n}={a}_{n - 1}+d \\text{ for } n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for nth term of an arithmetic sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum of the first [latex]n[\/latex]<br \/>\nterms of an arithmetic series<\/td>\n<td>[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.<\/li>\n<li>The constant between two consecutive terms is called the common difference.<\/li>\n<li>The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.<\/li>\n<li>The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.<\/li>\n<li>A recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\\ge 2[\/latex].<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\n<li>An explicit formula for an arithmetic sequence with common difference [latex]d[\/latex] is given by [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<li>An explicit formula can be used to find the number of terms in a sequence.<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[\/latex].<\/li>\n<li>The sum of the terms in a sequence is called a series.<\/li>\n<li>A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.<\/li>\n<li>The sum of the terms in an arithmetic sequence is called an arithmetic series.<\/li>\n<li>The sum of the first [latex]n[\/latex] terms of an arithmetic series can be found using a formula.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>arithmetic sequence<\/strong> a sequence in which the difference between any two consecutive terms is a constant<\/p>\n<p><strong>common difference<\/strong> the difference between any two consecutive terms in an arithmetic sequence<\/p>\n<p><strong>arithmetic series<\/strong> the sum of the terms in an arithmetic sequence<\/p>\n<p><strong>nth partial sum<\/strong> the sum of the first [latex]n[\/latex] terms of a sequence<\/p>\n<p><strong>series<\/strong> the sum of the terms in a sequence<\/p>\n<p><strong>summation notation<\/strong> a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1859\">\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>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 5847, 5832. <strong>Authored by<\/strong>: Web-Work Rochester. <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>Question ID 23735. <strong>Authored by<\/strong>: Roy Shahbazian. <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>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>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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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 5847, 5832\",\"author\":\"Web-Work Rochester\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 23735\",\"author\":\"Roy Shahbazian\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 29759\",\"author\":\"McClure, Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"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 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