{"id":14790,"date":"2018-09-27T18:36:01","date_gmt":"2018-09-27T18:36:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-2\/"},"modified":"2019-09-09T21:30:38","modified_gmt":"2019-09-09T21:30:38","slug":"arithmetic-sequences-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-2\/","title":{"raw":"Arithmetic Sequences","rendered":"Arithmetic Sequences"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Give terms of an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe values of the truck in the example are said to 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183536\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" \/>\r\n\r\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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183538\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Arithmetic Sequence<\/h3>\r\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:\r\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding Common Differences<\/h3>\r\nIs each sequence arithmetic? If so, find the common difference.\r\n<ol>\r\n \t<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\r\n \t<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"815179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815179\"]\r\n\r\nSubtract each term from the subsequent term to determine whether a common difference exists.\r\n<ol>\r\n \t<li>The sequence is not arithmetic because there is no common difference.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183540\/Eqn12.jpg\" alt=\"2 minus 1 = 1. 4 minus 2 = 2. 8 minus 4 = 4. 16 minus 8 equals 8.\" width=\"475\" height=\"27\" \/><\/li>\r\n \t<li>The sequence is arithmetic because there is a common difference. The common difference is 4.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183542\/Eqn22.jpg\" alt=\"1 minus negative 3 equals 4. 5 minus 1 equals 4. 9 minus 5 equals 4. 13 minus 9 equals 4.\" width=\"505\" height=\"27\" \/><\/li>\r\n<\/ol>\r\n<h4>Analysis of the Solution<\/h4>\r\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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183545\/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\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>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?<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the given sequence arithmetic? If so, find the common difference.\r\n<p style=\"text-align: center;\">[latex]\\left\\{18,\\text{ }16,\\text{ }14,\\text{ }12,\\text{ }10,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"157343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157343\"]\r\n\r\nThe sequence is arithmetic. The common difference is [latex]-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the given sequence arithmetic? If so, find the common difference.\r\n<p style=\"text-align: center;\">[latex]\\left\\{1,\\text{ }3,\\text{ }6,\\text{ }10,\\text{ }15,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"151399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151399\"]\r\n\r\nThe sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]172223[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Writing Terms of Arithmetic Sequences<\/h2>\r\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.\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\r\n<ol>\r\n \t<li>Add the common difference to the first term to find the second term.<\/li>\r\n \t<li>Add the common difference to the second term to find the third term.<\/li>\r\n \t<li>Continue until all of the desired terms are identified.<\/li>\r\n \t<li>Write the terms separated by commas within brackets.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Writing Terms of Arithmetic Sequences<\/h3>\r\nWrite the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].\r\n\r\n[reveal-answer q=\"223260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223260\"]\r\n\r\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.\r\n\r\nThe first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]\r\n<h3>Analysis of the Solution<\/h3>\r\nAs expected, the graph of the sequence consists of points on a line as shown in Figure 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183547\/CNX_Precalc_Figure_11_02_0042.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"487\" height=\"250\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nList the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .\r\n\r\n[reveal-answer q=\"531419\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"531419\"]\r\n\r\n[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]79151[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given any the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\r\n<ol>\r\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>\r\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>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Writing Terms of Arithmetic Sequences<\/h3>\r\nGiven [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex].\r\n\r\n[reveal-answer q=\"473118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473118\"]\r\n\r\nThe sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left\\{8,8+d,8+2d,8+3d,\\dots\\right\\}[\/latex]<\/p>\r\nWe know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex].\r\n\r\nWe can find the common difference [latex]d[\/latex].\r\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>\r\nFind the fifth term by adding the common difference to the fourth term.\r\n<p style=\"text-align: center;\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .\r\n\r\n[reveal-answer q=\"797232\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797232\"]\r\n\r\n[latex]{a}_{2}=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Using Formulas for Arithmetic Sequences<\/h2>\r\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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{n - 1}+d&amp;&amp; n\\ge 2 \\end{align}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Recursive Formula for an Arithmetic Sequence<\/h3>\r\nThe recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{n - 1}+d&amp;&amp; n\\ge 2 \\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an arithmetic sequence, write its recursive formula.<\/h3>\r\n<ol>\r\n \t<li>Subtract any term from the subsequent term to find the common difference.<\/li>\r\n \t<li>State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Writing a Recursive Formula for an Arithmetic Sequence<\/h3>\r\nWrite a <strong>recursive formula<\/strong> for the\u00a0<strong>arithmetic sequence<\/strong>.\r\n<p style=\"text-align: center;\">[latex]\\left\\{-18,-7,4,15\\,26,\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"366526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"366526\"]\r\n\r\nThe first term is given as [latex]-18[\/latex] . The common difference can be found by subtracting the first term from the second term.\r\n<p style=\"text-align: center;\">[latex]d=-7-\\left(-18\\right)=11[\/latex]<\/p>\r\nSubstitute the initial term and the common difference into the recursive formula for arithmetic sequences.\r\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>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183549\/CNX_Precalc_Figure_11_02_0052.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"250\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Do we have to subtract the first term from the second term to find the common difference?<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite a recursive formula for the arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{25\\text{, } 37\\text{, } 49\\text{, } 61\\text{, } \\dots\\right\\}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"110196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"110196\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{align}&amp;{a}_{1}=25\\\\ &amp;{a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<h2>Using Explicit Formulas for Arithmetic Sequences<\/h2>\r\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.\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/div>\r\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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183551\/CNX_Precalc_Figure_11_02_0062.jpg\" alt=\"A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.\" \/>\r\n\r\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. The graph is shown in Figure 4.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183553\/CNX_Precalc_Figure_11_02_0072.jpg\" alt=\"Graph of the arithmetic sequence. The points form a negative line.\" width=\"731\" height=\"250\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\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:\r\n<div style=\"text-align: center;\">[latex]{a}_{n}=-50n+250[\/latex]<\/div>\r\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].\r\n<div class=\"textbox\">\r\n<h3>A General Note: Explicit Formula for an Arithmetic Sequence<\/h3>\r\nAn explicit formula for the [latex]n\\text{th}[\/latex] term of an arithmetic sequence is given by\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/h3>\r\n<ol>\r\n \t<li>Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\r\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Writing the <em>n<\/em>th Term Explicit Formula for an Arithmetic Sequence<\/h3>\r\nWrite an explicit formula for the arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"360705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"360705\"]\r\n\r\nThe common difference can be found by subtracting the first term from the second term.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}d\\hfill &amp; ={a}_{2}-{a}_{1} \\\\ &amp; =12 - 2 \\\\ &amp;=10 \\end{align}[\/latex]<\/p>\r\nThe common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.\r\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>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of this sequence, represented in Figure 5, shows a slope of 10 and a vertical intercept of [latex]-8[\/latex] .\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183555\/CNX_Precalc_Figure_11_02_0082.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"276\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite an explicit formula for the following arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"42399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42399\"]\r\n\r\n[latex]{a}_{n}=53 - 3n[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]172280[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding the Number of Terms in a Finite Arithmetic Sequence<\/h2>\r\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.\r\n<div class=\"textbox\">\r\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>\r\n<ol>\r\n \t<li>Find the common difference [latex]d[\/latex].<\/li>\r\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\r\n \t<li>Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence<\/h3>\r\nFind the number of terms in the <strong>finite arithmetic sequence<\/strong>.\r\n<p style=\"text-align: center;\">[latex]\\left\\{8\\text{, }1\\text{, }-6\\text{, }\\dots\\text{, }-41\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"672670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"672670\"]\r\n\r\nThe common difference can be found by subtracting the first term from the second term.\r\n<p style=\"text-align: center;\">[latex]1 - 8=-7[\/latex]<\/p>\r\nThe common difference is [latex]-7[\/latex] . Substitute the common difference and the initial term of the sequence into the [latex]n\\text{th}[\/latex] term formula and simplify.\r\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>\r\nSubstitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}-41&amp;=15 - 7n \\\\ 8&amp;=n \\end{align}[\/latex]<\/p>\r\nThere are eight terms in the sequence.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the number of terms in the finite arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{6,11,16,\\dots,56\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"12633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"12633\"]\r\n\r\nThere are 11 terms in the sequence.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving Application Problems with Arithmetic Sequences<\/h2>\r\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:\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Solving Application Problems with Arithmetic Sequences<\/h3>\r\nA five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.\r\n<ol>\r\n \t<li>Write a formula for the child\u2019s weekly allowance in a given year.<\/li>\r\n \t<li>What will the child\u2019s allowance be when he is 16 years old?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"502685\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502685\"]\r\n<ol>\r\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:\r\n<div style=\"text-align: center;\">[latex]{A}_{n}=1+2n[\/latex]<\/div><\/li>\r\n \t<li>We can find the number of years since age 5 by subtracting.\r\n<div style=\"text-align: center;\">[latex]16 - 5=11[\/latex]<\/div>\r\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.\r\n<div style=\"text-align: center;\">[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<\/div>\r\nThe child\u2019s allowance at age 16 will be $23 per week.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\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?\r\n\r\n[reveal-answer q=\"433471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"433471\"]\r\n\r\nThe formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>recursive formula for nth term of an arithmetic sequence<\/td>\r\n<td>[latex]{a}_{n}={a}_{n - 1}+d\\phantom{\\rule{1}{0ex}}n\\ge 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>explicit formula for nth term of an arithmetic sequence<\/td>\r\n<td>[latex]\\begin{array}{l}{a}_{n}={a}_{1}+d\\left(n - 1\\right)\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.<\/li>\r\n \t<li>The constant between two consecutive terms is called the common difference.<\/li>\r\n \t<li>The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.<\/li>\r\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>\r\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>\r\n \t<li>As with any recursive formula, the initial term of the sequence must be given.<\/li>\r\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>\r\n \t<li>An explicit formula can be used to find the number of terms in a sequence.<\/li>\r\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137415238\" class=\"definition\">\r\n \t<dt>arithmetic sequence<\/dt>\r\n \t<dd id=\"fs-id1165137415244\">a sequence in which the difference between any two consecutive terms is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137415248\" class=\"definition\">\r\n \t<dt>common difference<\/dt>\r\n \t<dd id=\"fs-id1165135174993\">the difference between any two consecutive terms in an arithmetic sequence<\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Give terms of an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\n<\/ul>\n<\/div>\n<p>The values of the truck in the example are said to 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\/3675\/2018\/09\/27183536\/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\/3675\/2018\/09\/27183538\/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:<\/p>\n<p style=\"text-align: center;\">[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 shaded\">\n<h3>Example 1: 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=\"q815179\">Show Solution<\/span><\/p>\n<div id=\"q815179\" 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.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183540\/Eqn12.jpg\" alt=\"2 minus 1 = 1. 4 minus 2 = 2. 8 minus 4 = 4. 16 minus 8 equals 8.\" width=\"475\" height=\"27\" \/><\/li>\n<li>The sequence is arithmetic because there is a common difference. The common difference is 4.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183542\/Eqn22.jpg\" alt=\"1 minus negative 3 equals 4. 5 minus 1 equals 4. 9 minus 5 equals 4. 13 minus 9 equals 4.\" width=\"505\" height=\"27\" \/><\/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<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183545\/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<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>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?<\/h3>\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=\"bcc-box bcc-success\">\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,\\text{ }16,\\text{ }14,\\text{ }12,\\text{ }10,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157343\">Show Solution<\/span><\/p>\n<div id=\"q157343\" 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<\/div>\n<div class=\"bcc-box bcc-success\">\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,\\text{ }3,\\text{ }6,\\text{ }10,\\text{ }15,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q151399\">Show Solution<\/span><\/p>\n<div id=\"q151399\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172223\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172223&theme=oea&iframe_resize_id=ohm172223\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Writing Terms of Arithmetic Sequences<\/h2>\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<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/div>\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 shaded\">\n<h3>Example 2: 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=\"q223260\">Show Solution<\/span><\/p>\n<div id=\"q223260\" 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<h3>Analysis of the Solution<\/h3>\n<p>As expected, the graph of the sequence consists of points on a line as shown in Figure 2.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183547\/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<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\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=\"q531419\">Show Solution<\/span><\/p>\n<div id=\"q531419\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm79151\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79151&theme=oea&iframe_resize_id=ohm79151\" width=\"100%\" height=\"150\"><\/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 shaded\">\n<h3>Example 3: 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=\"q473118\">Show Solution<\/span><\/p>\n<div id=\"q473118\" 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,\\dots\\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=\"bcc-box bcc-success\">\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=\"q797232\">Show Solution<\/span><\/p>\n<div id=\"q797232\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Using Formulas for Arithmetic Sequences<\/h2>\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<div style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{n - 1}+d&& n\\ge 2 \\end{align}[\/latex]<\/div>\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<div style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{n - 1}+d&& n\\ge 2 \\end{align}[\/latex]<\/div>\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 shaded\">\n<h3>Example 4: Writing a Recursive Formula for an Arithmetic Sequence<\/h3>\n<p>Write a <strong>recursive formula<\/strong> for the\u00a0<strong>arithmetic sequence<\/strong>.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{-18,-7,4,15\\,26,\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366526\">Show Solution<\/span><\/p>\n<div id=\"q366526\" 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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183549\/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<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\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=\"bcc-box bcc-success\">\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\\text{, } 37\\text{, } 49\\text{, } 61\\text{, } \\dots\\right\\}[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q110196\">Show Solution<\/span><\/p>\n<div id=\"q110196\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{align}&{a}_{1}=25\\\\ &{a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\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<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/div>\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\/3675\/2018\/09\/27183551\/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. The graph is shown in Figure 4.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183553\/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 class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\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<div style=\"text-align: center;\">[latex]{a}_{n}=-50n+250[\/latex]<\/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<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/div>\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 shaded\">\n<h3>Example 5: 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 style=\"text-align: center;\">[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q360705\">Show Solution<\/span><\/p>\n<div id=\"q360705\" 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\\hfill & ={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, represented in Figure 5, shows a slope of 10 and a vertical intercept of [latex]-8[\/latex] .<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183555\/CNX_Precalc_Figure_11_02_0082.jpg\" alt=\"Graph of the arithmetic sequence. The points form a positive line.\" width=\"487\" height=\"276\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write an explicit formula for the following arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[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=\"q42399\">Show Solution<\/span><\/p>\n<div id=\"q42399\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=53 - 3n[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172280\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172280&theme=oea&iframe_resize_id=ohm172280\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Number of Terms in a Finite 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 shaded\">\n<h3>Example 6: 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>.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{8\\text{, }1\\text{, }-6\\text{, }\\dots\\text{, }-41\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q672670\">Show Solution<\/span><\/p>\n<div id=\"q672670\" 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 [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=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the number of terms in the finite arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{6,11,16,\\dots,56\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q12633\">Show Solution<\/span><\/p>\n<div id=\"q12633\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 11 terms in the sequence.<\/p>\n<\/div>\n<\/div>\n<\/div>\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:<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: 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=\"q502685\">Show Solution<\/span><\/p>\n<div id=\"q502685\" 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:\n<div style=\"text-align: center;\">[latex]{A}_{n}=1+2n[\/latex]<\/div>\n<\/li>\n<li>We can find the number of years since age 5 by subtracting.\n<div style=\"text-align: center;\">[latex]16 - 5=11[\/latex]<\/div>\n<p>We are looking for the child\u2019s allowance after 11 years. Substitute 11 into the formula to find the child\u2019s allowance at age 16.<\/p>\n<div style=\"text-align: center;\">[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<\/div>\n<p>The child\u2019s allowance at age 16 will be $23 per week.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\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=\"q433471\">Show Solution<\/span><\/p>\n<div id=\"q433471\" 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<\/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\\phantom{\\rule{1}{0ex}}n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for nth term of an arithmetic sequence<\/td>\n<td>[latex]\\begin{array}{l}{a}_{n}={a}_{1}+d\\left(n - 1\\right)\\end{array}[\/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<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137415238\" class=\"definition\">\n<dt>arithmetic sequence<\/dt>\n<dd id=\"fs-id1165137415244\">a sequence in which the difference between any two consecutive terms is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137415248\" class=\"definition\">\n<dt>common difference<\/dt>\n<dd id=\"fs-id1165135174993\">the difference between any two consecutive terms in an arithmetic sequence<\/dd>\n<\/dl>\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-14790\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14790","chapter","type-chapter","status-publish","hentry"],"part":14758,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14790\/revisions"}],"predecessor-version":[{"id":15856,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14790\/revisions\/15856"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14758"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14790\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=14790"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=14790"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=14790"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=14790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}