{"id":438,"date":"2019-07-15T22:45:02","date_gmt":"2019-07-15T22:45:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/sequences-defined-by-a-recursive-formula\/"},"modified":"2019-07-15T22:45:02","modified_gmt":"2019-07-15T22:45:02","slug":"sequences-defined-by-a-recursive-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/sequences-defined-by-a-recursive-formula\/","title":{"raw":"Sequences Defined by a Recursive Formula","rendered":"Sequences Defined by a Recursive Formula"},"content":{"raw":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Write terms of a sequence defined by a recursive formula<\/li><li>Write terms of a sequence using factorial notation<\/li><\/ul><\/div>\n\nSequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,\u2026. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.\n\nEach term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a <strong>recursive formula<\/strong>, a formula that defines the terms of a sequence using previous terms.\n\nA recursive formula always has two parts: the value of an initial term (or terms), and an equation defining [latex]{a}_{n}[\/latex] in terms of preceding terms. For example, suppose we know the following:\n\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&amp;=3 \\\\ {a}_{n}&amp;=2{a}_{n - 1}-1, \\text{for } n\\ge 2 \\end{align}[\/latex]\n\nWe can find the subsequent terms of the sequence using the first term.\n\n<\/p><p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&amp;=3\\\\ {a}_{2}&amp;=2{a}_{1}-1=2\\left(3\\right)-1=5\\\\ {a}_{3}&amp;=2{a}_{2}-1=2\\left(5\\right)-1=9\\\\ {a}_{4}&amp;=2{a}_{3}-1=2\\left(9\\right)-1=17\\end{align}[\/latex]\n\nSo the first four terms of the sequence are [latex]\\left\\{3,5,9,17\\right\\}[\/latex].\n\nThe recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.\n\n<\/p><p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&amp;=1 \\\\ {a}_{2}&amp;=1 \\\\ {a}_{n}&amp;={a}_{n - 1}+{a}_{n - 2}, \\text{for } n\\ge 3 \\end{align}[\/latex]\n\nTo find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We saw above that the eighth and ninth terms are 21 and 34, so\n\n<\/p><p style=\"text-align: center\">[latex]{a}_{10}={a}_{9}+{a}_{8}=34+21=55[\/latex]\n\n<\/p><div class=\"textbox\"><h3>A General Note: Recursive Formula<\/h3>A <strong>recursive formula<\/strong> is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><h4>Must the first two terms always be given in a recursive formula?<\/h4><em>No. The Fibonacci sequence defines each term using the two preceding terms, but many recursive formulas define each term using only one preceding term. These sequences need only the first term to be defined.<\/em>\n\n<\/div><div class=\"textbox\"><h3>How To: Given a recursive formula with only the first term provided, write the first [latex]n[\/latex] terms of a sequence.<\/h3><ol><li>Identify the initial term, [latex]{a}_{1}[\/latex], which is given as part of the formula. This is the first term.<\/li><li>To find the second term, [latex]{a}_{2}[\/latex], substitute the initial term into the formula for [latex]{a}_{n - 1}[\/latex]. Solve.<\/li><li>To find the third term, [latex]{a}_{3}[\/latex], substitute the second term into the formula. Solve.<\/li><li>Repeat until you have solved for the [latex]n\\text{th}[\/latex] term.<\/li><\/ol><\/div><div class=\"textbox exercises\"><h3>Example: Writing the Terms of a Sequence Defined by a Recursive Formula<\/h3>Write the first five terms of the sequence defined by the recursive formula.\n\n<p style=\"text-align: center\">[latex]\\begin{align} {a}_{1}&amp;=9 \\\\ {a}_{n}&amp;=3{a}_{n - 1}-20\\text{, for }n\\ge 2 \\end{align}[\/latex]\n\n[reveal-answer q=\"748916\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"748916\"]\n\nThe first term is given in the formula. For each subsequent term, we replace [latex]{a}_{n - 1}[\/latex] with the value of the preceding term.\n\n<\/p><p style=\"text-align: center\">[latex]\\begin{align}&amp;n=1 &amp;&amp; {a}_{1}=9 \\\\ &amp;n=2 &amp;&amp; {a}_{2}=3{a}_{1}-20=3\\left(9\\right)-20=27 - 20=7 \\\\ &amp;n=3 &amp;&amp; {a}_{3}=3{a}_{2}-20=3\\left(7\\right)-20=21 - 20=1 \\\\ &amp;n=4 &amp;&amp; {a}_{4}=3{a}_{3}-20=3\\left(1\\right)-20=3 - 20=-17 \\\\ &amp;n=5 &amp;&amp; {a}_{5}=3{a}_{4}-20=3\\left(-17\\right)-20=-51 - 20=-71 \\end{align}[\/latex]\n\nThe first five terms are [latex]\\left\\{9,7,1,-17,-71\\right\\}[\/latex]\n\n[\/hidden-answer]\n\n<\/p><\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Write the first five terms of the sequence defined by the recursive formula.\n\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&amp;=2\\\\ {a}_{n}&amp;=2{a}_{n - 1}+1\\text{, for }n\\ge 2\\end{align}[\/latex]\n\n[reveal-answer q=\"378600\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"378600\"]\n\n[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5812&amp;theme=oea&amp;iframe_resize_id=mom21[\/embed]\n\n\n\n<\/p><\/div><h2>Using Factorial Notation<\/h2><div class=\"textbox examples\"><h3>a new operator: factorial<\/h3>We saw a new operator introduced earlier while studying functions, the composition operator, [latex]\\circ[\/latex], which indicated we should compose two functions.\n\nNow we see another new operator, factorial, [latex]![\/latex]. It doesn't mean we should get excited about its pronunciation, though! Factorial asks us to multiply together all the positive integers that come before the number in front of the symbol. See the examples below.\n\nFactorial is used heavily in Combinatorics, a branch of mathematics that is concerned with methods for counting sets.\n\n<\/div>The formulas for some sequences include products of consecutive positive integers. <strong>[latex]n[\/latex] factorial<\/strong>, written as [latex]n![\/latex], is the product of the positive integers from 1 to [latex]n[\/latex]. For example,\n\n<div style=\"text-align: center\">[latex]\\begin{align}4!&amp;=4\\cdot 3\\cdot 2\\cdot 1=24 \\\\ 5!&amp;=5\\cdot 4\\cdot 3\\cdot 2\\cdot 1=120\\\\ \\text{ } \\end{align}[\/latex]<\/div><div><\/div>An example of formula containing a <strong>factorial<\/strong> is [latex]{a}_{n}=\\left(n+1\\right)![\/latex]. The sixth term of the sequence can be found by substituting 6 for [latex]n[\/latex].\n\n<div style=\"text-align: center\">[latex]\\begin{align}{a}_{6}=\\left(6+1\\right)!=7!=7\\cdot 6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1=5040 \\\\ \\text{ }\\end{align}[\/latex]<\/div><div><\/div>The factorial of any whole number [latex]n[\/latex] is [latex]n\\left(n - 1\\right)![\/latex] We can therefore also think of [latex]5![\/latex] as [latex]5\\cdot 4!\\text{.}[\/latex]\n\n<div class=\"textbox\"><h3>A GENERAL NOTE: FACTORIAL<\/h3><strong><em>n<\/em> factorial<\/strong> is a mathematical operation that can be defined using a recursive formula. The factorial of [latex]n[\/latex], denoted [latex]n![\/latex], is defined for a positive integer [latex]n[\/latex] as:\n\n<div style=\"text-align: center\">[latex]\\begin{array}{l}0!=1\\\\ 1!=1\\\\ n!=n\\left(n - 1\\right)\\left(n - 2\\right)\\cdots \\left(2\\right)\\left(1\\right)\\text{, for }n\\ge 2\\end{array}[\/latex]<\/div>The special case [latex]0![\/latex] is defined as [latex]0!=1[\/latex].\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61071&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><h3>CAN FACTORIALS ALWAYS BE FOUND USING A CALCULATOR?<\/h3><em>No. Factorials get large very quickly\u2014faster than even exponential functions! When the output gets too large for the calculator, it will not be able to calculate the factorial.<\/em>\n\n<\/div><div class=\"textbox exercises\"><h3>EXAMPLE: WRITING THE TERMS OF A SEQUENCE USING FACTORIALS<\/h3>Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{5n}{\\left(n+2\\right)!}[\/latex].\n[reveal-answer q=\"443745\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"443745\"]\n\nSubstitute [latex]n=1,n=2[\/latex], and so on in the formula.\n\n<div style=\"text-align: center\">[latex]\\begin{align}&amp;n=1 &amp;&amp; {a}_{1}=\\dfrac{5\\left(1\\right)}{\\left(1+2\\right)!}=\\dfrac{5}{3!}=\\dfrac{5}{3\\cdot 2\\cdot 1}=\\dfrac{5}{6} \\\\[1mm] &amp;n=2 &amp;&amp; {a}_{2}=\\dfrac{5\\left(2\\right)}{\\left(2+2\\right)!}=\\dfrac{10}{4!}=\\dfrac{10}{4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{5}{12} \\\\[1mm] &amp;n=3 &amp;&amp; {a}_{3}=\\dfrac{5\\left(3\\right)}{\\left(3+2\\right)!}=\\dfrac{15}{5!}=\\dfrac{15}{5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{1}{8} \\\\[1mm] &amp;n=4 &amp;&amp; {a}_{4}=\\dfrac{5\\left(4\\right)}{\\left(4+2\\right)!}=\\dfrac{20}{6!}=\\dfrac{20}{6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{1}{36} \\\\[1mm] &amp;n=5 &amp;&amp; {a}_{5}=\\dfrac{5\\left(5\\right)}{\\left(5+2\\right)!}=\\dfrac{25}{7!}=\\dfrac{25}{7\\cdot 6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{5}{1\\text{,}008}\\\\ \\text{ } \\end{align}[\/latex]<\/div><div><\/div>The first five terms are [latex]\\left\\{\\dfrac{5}{6},\\dfrac{5}{12},\\dfrac{1}{8},\\dfrac{1}{36},\\dfrac{5}{1,008}\\right\\}[\/latex].\n\n<strong>Analysis of the Solution<\/strong>\n\nThe figure below shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the factorial term in the denominator results in the denominator becoming much larger than the numerator as [latex]n[\/latex] increases. This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202504\/CNX_Precalc_Figure_11_01_0082.jpg\" alt=\"Graph of a scatter plot with labeled points: (1, 5\/6), (2, 5\/12), (3, 1\/8), (4, 1\/36), and (5, 5\/1008). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"487\" height=\"284\">\n\n[\/hidden-answer]\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{\\left(n+1\\right)!}{2n}[\/latex].\n[reveal-answer q=\"953785\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"953785\"]\n\nThe first five terms are [latex]\\displaystyle \\left\\{1, \\frac{3}{2}, 4,15,72\\right\\}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68773&amp;theme=oea&amp;iframe_resize_id=mom20[\/embed]\n\n\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write terms of a sequence defined by a recursive formula<\/li>\n<li>Write terms of a sequence using factorial notation<\/li>\n<\/ul>\n<\/div>\n<p>Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,\u2026. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.<\/p>\n<p>Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a <strong>recursive formula<\/strong>, a formula that defines the terms of a sequence using previous terms.<\/p>\n<p>A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining [latex]{a}_{n}[\/latex] in terms of preceding terms. For example, suppose we know the following:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&=3 \\\\ {a}_{n}&=2{a}_{n - 1}-1, \\text{for } n\\ge 2 \\end{align}[\/latex]<\/p>\n<p>We can find the subsequent terms of the sequence using the first term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&=3\\\\ {a}_{2}&=2{a}_{1}-1=2\\left(3\\right)-1=5\\\\ {a}_{3}&=2{a}_{2}-1=2\\left(5\\right)-1=9\\\\ {a}_{4}&=2{a}_{3}-1=2\\left(9\\right)-1=17\\end{align}[\/latex]<\/p>\n<p>So the first four terms of the sequence are [latex]\\left\\{3,5,9,17\\right\\}[\/latex].<\/p>\n<p>The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&=1 \\\\ {a}_{2}&=1 \\\\ {a}_{n}&={a}_{n - 1}+{a}_{n - 2}, \\text{for } n\\ge 3 \\end{align}[\/latex]<\/p>\n<p>To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We saw above that the eighth and ninth terms are 21 and 34, so<\/p>\n<p style=\"text-align: center\">[latex]{a}_{10}={a}_{9}+{a}_{8}=34+21=55[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Recursive Formula<\/h3>\n<p>A <strong>recursive formula<\/strong> is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Must the first two terms always be given in a recursive formula?<\/h4>\n<p><em>No. The Fibonacci sequence defines each term using the two preceding terms, but many recursive formulas define each term using only one preceding term. These sequences need only the first term to be defined.<\/em><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a recursive formula with only the first term provided, write the first [latex]n[\/latex] terms of a sequence.<\/h3>\n<ol>\n<li>Identify the initial term, [latex]{a}_{1}[\/latex], which is given as part of the formula. This is the first term.<\/li>\n<li>To find the second term, [latex]{a}_{2}[\/latex], substitute the initial term into the formula for [latex]{a}_{n - 1}[\/latex]. Solve.<\/li>\n<li>To find the third term, [latex]{a}_{3}[\/latex], substitute the second term into the formula. Solve.<\/li>\n<li>Repeat until you have solved for the [latex]n\\text{th}[\/latex] term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Terms of a Sequence Defined by a Recursive Formula<\/h3>\n<p>Write the first five terms of the sequence defined by the recursive formula.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} {a}_{1}&=9 \\\\ {a}_{n}&=3{a}_{n - 1}-20\\text{, for }n\\ge 2 \\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q748916\">Show Solution<\/span><\/p>\n<div id=\"q748916\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is given in the formula. For each subsequent term, we replace [latex]{a}_{n - 1}[\/latex] with the value of the preceding term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&n=1 && {a}_{1}=9 \\\\ &n=2 && {a}_{2}=3{a}_{1}-20=3\\left(9\\right)-20=27 - 20=7 \\\\ &n=3 && {a}_{3}=3{a}_{2}-20=3\\left(7\\right)-20=21 - 20=1 \\\\ &n=4 && {a}_{4}=3{a}_{3}-20=3\\left(1\\right)-20=3 - 20=-17 \\\\ &n=5 && {a}_{5}=3{a}_{4}-20=3\\left(-17\\right)-20=-51 - 20=-71 \\end{align}[\/latex]<\/p>\n<p>The first five terms are [latex]\\left\\{9,7,1,-17,-71\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the first five terms of the sequence defined by the recursive formula.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}{a}_{1}&=2\\\\ {a}_{n}&=2{a}_{n - 1}+1\\text{, for }n\\ge 2\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q378600\">Show Solution<\/span><\/p>\n<div id=\"q378600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{2, 5, 11, 23, 47\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm5812\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5812&#38;theme=oea&#38;iframe_resize_id=ohm5812&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Factorial Notation<\/h2>\n<div class=\"textbox examples\">\n<h3>a new operator: factorial<\/h3>\n<p>We saw a new operator introduced earlier while studying functions, the composition operator, [latex]\\circ[\/latex], which indicated we should compose two functions.<\/p>\n<p>Now we see another new operator, factorial, [latex]![\/latex]. It doesn&#8217;t mean we should get excited about its pronunciation, though! Factorial asks us to multiply together all the positive integers that come before the number in front of the symbol. See the examples below.<\/p>\n<p>Factorial is used heavily in Combinatorics, a branch of mathematics that is concerned with methods for counting sets.<\/p>\n<\/div>\n<p>The formulas for some sequences include products of consecutive positive integers. <strong>[latex]n[\/latex] factorial<\/strong>, written as [latex]n![\/latex], is the product of the positive integers from 1 to [latex]n[\/latex]. For example,<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}4!&=4\\cdot 3\\cdot 2\\cdot 1=24 \\\\ 5!&=5\\cdot 4\\cdot 3\\cdot 2\\cdot 1=120\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<p>An example of formula containing a <strong>factorial<\/strong> is [latex]{a}_{n}=\\left(n+1\\right)![\/latex]. The sixth term of the sequence can be found by substituting 6 for [latex]n[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}{a}_{6}=\\left(6+1\\right)!=7!=7\\cdot 6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1=5040 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div><\/div>\n<p>The factorial of any whole number [latex]n[\/latex] is [latex]n\\left(n - 1\\right)![\/latex] We can therefore also think of [latex]5![\/latex] as [latex]5\\cdot 4!\\text{.}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: FACTORIAL<\/h3>\n<p><strong><em>n<\/em> factorial<\/strong> is a mathematical operation that can be defined using a recursive formula. The factorial of [latex]n[\/latex], denoted [latex]n![\/latex], is defined for a positive integer [latex]n[\/latex] as:<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}{l}0!=1\\\\ 1!=1\\\\ n!=n\\left(n - 1\\right)\\left(n - 2\\right)\\cdots \\left(2\\right)\\left(1\\right)\\text{, for }n\\ge 2\\end{array}[\/latex]<\/div>\n<p>The special case [latex]0![\/latex] is defined as [latex]0!=1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm61071\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61071&#38;theme=oea&#38;iframe_resize_id=ohm61071&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>CAN FACTORIALS ALWAYS BE FOUND USING A CALCULATOR?<\/h3>\n<p><em>No. Factorials get large very quickly\u2014faster than even exponential functions! When the output gets too large for the calculator, it will not be able to calculate the factorial.<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE: WRITING THE TERMS OF A SEQUENCE USING FACTORIALS<\/h3>\n<p>Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{5n}{\\left(n+2\\right)!}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q443745\">Show Solution<\/span><\/p>\n<div id=\"q443745\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]n=1,n=2[\/latex], and so on in the formula.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}&n=1 && {a}_{1}=\\dfrac{5\\left(1\\right)}{\\left(1+2\\right)!}=\\dfrac{5}{3!}=\\dfrac{5}{3\\cdot 2\\cdot 1}=\\dfrac{5}{6} \\\\[1mm] &n=2 && {a}_{2}=\\dfrac{5\\left(2\\right)}{\\left(2+2\\right)!}=\\dfrac{10}{4!}=\\dfrac{10}{4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{5}{12} \\\\[1mm] &n=3 && {a}_{3}=\\dfrac{5\\left(3\\right)}{\\left(3+2\\right)!}=\\dfrac{15}{5!}=\\dfrac{15}{5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{1}{8} \\\\[1mm] &n=4 && {a}_{4}=\\dfrac{5\\left(4\\right)}{\\left(4+2\\right)!}=\\dfrac{20}{6!}=\\dfrac{20}{6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{1}{36} \\\\[1mm] &n=5 && {a}_{5}=\\dfrac{5\\left(5\\right)}{\\left(5+2\\right)!}=\\dfrac{25}{7!}=\\dfrac{25}{7\\cdot 6\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 1}=\\dfrac{5}{1\\text{,}008}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<p>The first five terms are [latex]\\left\\{\\dfrac{5}{6},\\dfrac{5}{12},\\dfrac{1}{8},\\dfrac{1}{36},\\dfrac{5}{1,008}\\right\\}[\/latex].<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>The figure below shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the factorial term in the denominator results in the denominator becoming much larger than the numerator as [latex]n[\/latex] increases. This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202504\/CNX_Precalc_Figure_11_01_0082.jpg\" alt=\"Graph of a scatter plot with labeled points: (1, 5\/6), (2, 5\/12), (3, 1\/8), (4, 1\/36), and (5, 5\/1008). The x-axis is labeled n and the y-axis is labeled a_n.\" width=\"487\" height=\"284\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\\dfrac{\\left(n+1\\right)!}{2n}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q953785\">Show Solution<\/span><\/p>\n<div id=\"q953785\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first five terms are [latex]\\displaystyle \\left\\{1, \\frac{3}{2}, 4,15,72\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm68773\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=68773&#38;theme=oea&#38;iframe_resize_id=ohm68773&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-438\">\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 5812. <strong>Authored by<\/strong>: David Lippman. <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 LicenseCC-BY + GPL<\/li><li>Question ID 6107 . <strong>Authored by<\/strong>: Gregg Harbaugh. <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 19458. <strong>Authored by<\/strong>: James Sousa. <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 5846. <strong>Authored by<\/strong>: WebWork-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 68773. <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><\/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":17533,"menu_order":7,"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 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