{"id":89032,"date":"2021-08-13T10:56:29","date_gmt":"2021-08-13T15:56:29","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=89032"},"modified":"2026-03-28T10:39:07","modified_gmt":"2026-03-28T15:39:07","slug":"geometric-sequences","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/geometric-sequences\/","title":{"rendered":"Geometric Sequences"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_-3BOIFB7TLs\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_-3BOIFB7TLs\" data-source-videoID=\"-3BOIFB7TLs\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Geometric Sequences Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Geometric Sequences\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_-3BOIFB7TLs:hover {cursor:pointer;} img#videoThumbnailImage_-3BOIFB7TLs {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1725-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_-3BOIFB7TLs\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_-3BOIFB7TLs\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Geometric Sequences\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_-3BOIFB7TLs\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_-3BOIFB7TLs\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_-3BOIFB7TLs\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction VcY_Function() {\n  var x = document.getElementById(\"VcY\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"VcY_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"VcY\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Writing_Formulas_for_Geometric_Sequences\" class=\"smooth-scroll\">Writing Formulas for Geometric Sequences<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Finding_Terms_in_Geometric_Sequences\" class=\"smooth-scroll\">Finding Terms in Geometric Sequences<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Video_Transcript\" class=\"smooth-scroll\">Video Transcript<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Geometric_Sequence_Examples\" class=\"smooth-scroll\">Geometric Sequence Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Geometric_Series\" class=\"smooth-scroll\">Geometric Series<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Geometric_Sequence_Practice_Questions\" class=\"smooth-scroll\">Geometric Sequence Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"overview\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"overview\">Overview<\/label><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"overview-spoiler\">\n<h2><span id=\"Writing_Formulas_for_Geometric_Sequences\" class=\"m-toc-anchor\"><\/span>Writing Formulas for Geometric Sequences<\/h2>\n<p>A <strong>geometric sequence<\/strong> is a list of numbers, where the next term of the sequence is found by multiplying the term by a constant, called the common ratio.<\/p>\n<p>The general form of the geometric sequence formula is: \\(a_n=a_1r^{(n-1)}\\), where \\(r\\) is the common ratio, \\(a_1\\) is the first term, and \\(n\\) is the placement of the term in the sequence.<\/p>\n<p>Here is a geometric sequence: \\(1,3,9,27,81,\u2026\\)<\/p>\n<p>To find the formula for this geometric sequence, start by determining the common ratio, which is \\(3\\), since the terms are increasing by a factor of \\(3\\). Then identify the first term, \\(a_1\\), which is \\(1\\). Therefore, the formula for this geometric sequence is \\(a_n=1\u00b73^{(n-1)}\\).<\/p>\n<p>Use the formula for the geometric sequence to find the \\(8^{th}\\) term:<\/p>\n<p style=\"text-align: center;\">\\(a_8=1\u00b73^{(8-1)}\\)<\/p>\n<p style=\"text-align: center;\">\\(a_8=2{,}187\\)<\/p>\n<h3><span id=\"Example\" class=\"m-toc-anchor\"><\/span>Example:<\/h3>\n<p>Here is a geometric sequence: \\(2,10,50,250,1{,}250,\u2026\\)<\/p>\n<p>Find the formula for the geometric sequence, then find the \\(7^{th}\\) term.<\/p>\n<p>Common ratio is \\(5\\), first term is \\(2\\), therefore the formula for the geometric sequence is, \\(a_n=2\u00b75^{(n-1)}\\).<\/p>\n<p style=\"text-align: center;\">\\(a_7=2\u00b75^{(7-1)}=31{,}250\\)<\/p>\n<h2><span id=\"Finding_Terms_in_Geometric_Sequences\" class=\"m-toc-anchor\"><\/span>Finding Terms in Geometric Sequences<\/h2>\n<h3><span id=\"What_is_a_geometric_sequence\" class=\"m-toc-anchor\"><\/span>What is a geometric sequence?<\/h3>\n<p>A geometric sequence is an ordered set of numbers in which each term is a fixed multiple of the number that comes before it. Geometric sequences use multiplication to find each subsequent term. Each term gets multiplied by a common ratio, resulting in the next term in the sequence. In the geometric sequence shown below, the common ratio is 2. In other words, each term is multiplied by 2. The resulting product is the next term in the sequence.<\/p>\n<p style=\"text-align: center;\">\\(1,2,4,8,16,32,64\u2026\\)<\/p>\n<h3><span id=\"How_do_you_find_a_term_in_a_geometric_sequence_when_given_a_formula\" class=\"m-toc-anchor\"><\/span>How do you find a term in a geometric sequence when given a formula?<\/h3>\n<p>The formula \\(a_n=a_1r^{(n-1)}\\) is used to identify any number in a given geometric sequence. In this formula, \\(n\\) stands for the number in the sequence that needs to be identified. \\(a_1\\) stands for the first term in the sequence, and \\(r\\) stands for the common ratio. Consider how this formula applies to the following geometric sequence:<\/p>\n<p style=\"text-align: center;\">\\(2,10,50,250\u2026\\)<\/p>\n<p>In this geometric sequence, the common ratio, or \\(r\\), equals \\(5\\). As the sequence progresses, each term is multiplied by \\(5\\). Use the formula given to identify the \\(6^{th}\\) term in this sequence.<\/p>\n<p>Since the variable \\(n\\) stands for the number in the sequence that needs to be identified, replace \\(n\\) with \\(6\\) in the formula. The variable \\(a_1\\) stands for the first term in the sequence, which is \\(2\\). The variable r represents the common ratio, so replace \\(r\\) with \\(5\\) in the formula:<\/p>\n<p style=\"text-align: center;\">\\(n=6\\)\\(a_1=2\\)\\(r=5\\)<\/p>\n<p>From here, rewrite the formula and substitute the variables with the numbers they represent:<\/p>\n<p style=\"text-align: center;\">\\(a_6=(2)(5)^{(6-1)}\\)<\/p>\n<p>Now that the variables are replaced with their corresponding values, solve the equation using the order of operations.<\/p>\n<p>First, simplify the exponents. Since \\(6-1=5\\), we can rewrite the equation using the exponent \\(5\\).<\/p>\n<p style=\"text-align: center;\">\\(a_6=(2)(5)^{(5)}\\)<\/p>\n<p>Next, simplify the exponent \\(5^5\\). \\(5^5=5\u00d75\u00d75\u00d75\u00d75\\), which equals \\(3{,}125\\). Rewrite the equation using \\(3{,}125\\).<\/p>\n<p style=\"text-align: center;\">\\(a_6=(10)(3{,}125)\\)<\/p>\n<p>From here, multiply \\(10\u00d73{,}125\\) to get the final answer. Since \\(10\u00d73{,}125=31{,}250\\), the \\(6^{th}\\) term in the geometric sequence is \\(31{,}250\\).<\/p>\n<p style=\"text-align: center;\">\\(a_6=31{,}250\\)<\/p>\n<h3><span id=\"How_do_you_find_a_term_in_a_geometric_sequence_without_a_formula\" class=\"m-toc-anchor\"><\/span>How do you find a term in a geometric sequence without a formula?<\/h3>\n<p>A missing term in a geometric sequence can be found without using a formula. Analyze the pattern and consider the common ratio to identify the missing term. Consider the following geometric sequence:<\/p>\n<p style=\"text-align: center;\">\\(10,30,90,270,\\) _____ \\(,2{,}430\u2026\\)<\/p>\n<p>First, analyze the pattern to identify the common ratio. Start with the first term, \\(10\\). \\(10\\) multiplied by what number results in a product of \\(30\\)? The correct answer is \\(3\\), so the common ratio is \\(3\\). Check the rest of the terms using this common ratio. \\(30\u00d73=90\\) and \\(90\u00d73=270\\).<\/p>\n<p>Next, multiply \\(270\\) by \\(3\\) to get the missing term in the geometric sequence. Since \\(270\u00d73=810\\), the missing term is \\(810\\).<\/p>\n<div id=\"pqs\">\u00a0<\/div>\n<p style=\"text-align: center;\">\\(10,30,90,270,\\mathbf{810},2{,}430\u2026\\)<\/p>\n<p>Check your work by multiplying \\(810\\) by \\(3\\). If correct, the product should be the subsequent term in the sequence, \\(2{,}450\\). Since \\(810\u00d73=2{,}430\\), the answer is correct.<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Video_Transcript\" class=\"m-toc-anchor\"><\/span>Video Transcript<\/h2>\n<p>Hello, and welcome to this video on geometric sequences! <\/p>\n<p>Today we\u2019ll explore how to find a term in a geometric sequence using a formula. We\u2019ll also learn how to find the sum of a specific number of terms in a geometric sequence. <\/p>\n<p>Before we dive in, let\u2019s review a few things. First, a <strong>sequence<\/strong> is an ordered set of numbers with a pattern. The pattern helps us predict what each term might be. For example, 2, 4, 6, 8, and 10 are a sequence. <\/p>\n<p>A <strong>term<\/strong> is an individual expression, or number, in the sequence.<\/p>\n<h2><span id=\"Geometric_Sequence_Examples\" class=\"m-toc-anchor\"><\/span>Geometric Sequence Examples<\/h2>\n<p>\nA <strong>geometric sequence<\/strong> is a special type of sequence. Each term is a fixed multiple of the number that comes before it. <\/p>\n<p>For example, let\u2019s say my first number is 2, and I multiply 2 by 5 to get 10. Then I multiply 10 by 5 to get 50. I can multiply 50 by 5 to get 250, and so on. <\/p>\n<div class=\"examplesentence\">\\(2, 10, 50, 250&#8230;\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn geometric sequences, we use multiplication to find each subsequent term. The number we multiply by is called the <strong>common ratio<\/strong>. Each term gets multiplied by the common ratio, resulting in the next term in the sequence. In this geometric sequence, the common ratio is 5. <\/p>\n<h3><span id=\"Finding_Numbers\" class=\"m-toc-anchor\"><\/span>Finding Numbers<\/h3>\n<p>\nTo find any number in a given sequence, we can use the following formula: <\/p>\n<div class=\"examplesentence\">\\(a_{n}=a_{1}r^{(n-1)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn this formula, <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> stands for the number in the sequence we are asked to find. <span style=\"font-style:normal; font-size:90%\">\\(a_{1}\\)<\/span> stands for the first term in the sequence, and <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span> stands for the common ratio. Let\u2019s look at how we can use this formula. Given the geometric sequence:<\/p>\n<div class=\"examplesentence\">\\(10, 30, 90, 270&#8230;\\)<\/div>\n<p>\n&nbsp;<br \/>\nwe can see that the common ratio, or <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span>, equals 3 because each time we\u2019re multiplying by 3. <\/p>\n<div class=\"examplesentence\">\\(10\\times 3=30\\)<br \/>\n\\(30\\times 3=90\\)<br \/>\n\\(90\\times 3=270\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd so on and so forth. <\/p>\n<p>Let\u2019s use our formula to find the 6th term in this sequence. <\/p>\n<p>Since we need to identify the 6th term, we can replace the variable <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> with 6. Remember that <span style=\"font-style:normal; font-size:90%\">\\(a_{1}\\)<\/span> always stands for the first term in our sequence. In this case, the first term is 10. Let\u2019s rewrite the formula and replace the variables with the numbers that they represent.<\/p>\n<p>So we\u2019re looking for the 6th term, so:<\/p>\n<div class=\"examplesentence\">\\(a_{6}=10(3)^{6-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow that we\u2019ve replaced the variables with numbers, we can solve the equation using the order of operations. First, we need to simplify the exponents. Since 6 &#8211; 1 = 5, we can rewrite the equation using one exponent.<\/p>\n<div class=\"examplesentence\">\\(a_{6}=10(3)^{5}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can solve <span style=\"font-style:normal; font-size:90%\">\\(3^{5}\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\(3\\times 3\\times 3\\times 3\\times 3\\)<\/span>, which equals 243.<\/p>\n<div class=\"examplesentence\">\\(a_{6}=10(243)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we can multiply <span style=\"font-style:normal; font-size:90%\">\\(10\\times 243\\)<\/span> to get the final answer. The 6th term in the geometric sequence is 2,430.<\/p>\n<h2><span id=\"Geometric_Series\" class=\"m-toc-anchor\"><\/span>Geometric Series<\/h2>\n<p>\nA <strong>geometric series<\/strong> is the sum of all terms in a geometric sequence. Let\u2019s consider the last sequence we looked at, which was 10, 30, 90, 270, and so on. <\/p>\n<p>As a geometric series, this is written as <span style=\"font-style:normal; font-size:90%\">\\(10+30+90+270+&#8230;\\)<\/span> <\/p>\n<p>To find the sum of a specific number of terms in a geometric sequence, we can use this formula:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{n}=\\frac{a_{1}(1-r^{n})}{1-r}\\)<\/div>\n<p>\n&nbsp;<br \/>\nIn this formula, <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> stands for the number of terms added together. Like the first formula we learned about, <span style=\"font-style:normal; font-size:90%\">\\(a_{1}\\)<\/span> stands for the first term in the sequence, and <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span> stands for the common ratio. <\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s walk through an example together. Using the same geometric sequence as our last example, let\u2019s find the sum of the first 8 terms: <\/p>\n<div class=\"examplesentence\">\\(10, 30, 90, 270&#8230;\\)<\/div>\n<p>\n&nbsp;<br \/>\nUsing this sequence and the formula <span style=\"font-style:normal; font-size:90%\">\\(s_{n}=\\frac{a_{1}(1-r^{n})}{1-r}\\)<\/span>, let\u2019s consider what each variable can be replaced with. <\/p>\n<p>Remember that <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> stands for the number of terms. Since we are looking for the sum of the first 8 terms, we can substitute 8 for <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span>. <span style=\"font-style:normal; font-size:90%\">\\(a_{1}\\)<\/span> stands for the first term in the sequence, which is 10. Recall that <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span> stands for the common ratio. The common ratio is 3, so we can substitute <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span> with 3 in our formula. <\/p>\n<p>Now, let\u2019s rewrite the equation and replace the variables with the numbers they represent:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{10(1-3^{8})}{1-3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow that we replaced the variables with numbers, we can solve the equation using the order of operations. First, we need to simplify the exponents. Since <span style=\"font-style:normal; font-size:90%\">\\(3^{8}=6,561\\)<\/span>, we can rewrite the equation using this product.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{10(1-6,561)}{1-3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can simplify the parentheses. Since <span style=\"font-style:normal; font-size:90%\">\\(1-6,561=-6,560\\)<\/span>, we can rewrite the equation again.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{10(-6,560)}{1-3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we can simplify the numerator and denominator.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{-65,600}{-2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd to get our final answer, we just divide. <span style=\"font-style:normal; font-size:90%\">\\(-65,600\\div -2=32,800\\)<\/span>. So the sum of our 8 terms is 32,800.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nGeometric sequences are patterns, and patterns are all around us. Knowing how they work will help you identify and use them in the real world. <\/p>\n<p>For example, let\u2019s say that you want to share a YouTube video with friends. You might start by messaging 2 friends about this. In your message, you might ask each friend to share the video with 4 other friends. That would mean 8 people were messaged, since <span style=\"font-style:normal; font-size:90%\">\\(2\\times 4=8\\)<\/span>. If each of those people shares with 4 more people, then how many people would have seen the video? 32, because <span style=\"font-style:normal; font-size:90%\">\\(8\\times 4=32\\)<\/span>. <\/p>\n<p>As you can see, we have a geometric pattern. 2, 8, 32, and so on. The common ratio is 4 since each person is asked to share the video with 4 friends. Let\u2019s find out how many people received the YouTube video after 8 rounds of messaging, including the original message shared with 2 friends. <\/p>\n<p>To solve this problem, we need to know the sum of the first 8 terms in the sequence. Remember which formula can help us find the sum. Our sum formula is:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{n}=\\frac{a_{1}(1-r^{n})}{1-r}\\)<\/div>\n<p>\n&nbsp;<br \/>\nUsing information from the problem, let\u2019s replace variables with the numbers they represent. <span style=\"font-style:normal; font-size:90%\">\\(n=8\\)<\/span> because we are finding the sum of the first 8 terms in the sequence. <span style=\"font-style:normal; font-size:90%\">\\(a_{1}=2\\)<\/span> since the first number in the sequence is 2. And <span style=\"font-style:normal; font-size:90%\">\\(r=4\\)<\/span> because the common ratio is 4. So we\u2019re gonna substitute in our variables.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{2(1-(4)^{8})}{1-4}\\)<\/div>\n<p>\n&nbsp;<br \/>\nFrom here, we can solve the equation using the order of operations. First, we need to simplify the exponent inside the parentheses. Since <span style=\"font-style:normal; font-size:90%\">\\(4^{8}=65,536\\)<\/span>, we can rewrite the equation without an exponent.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{2(1-65,536)}{1-4}\\)<\/div>\n<p>\n&nbsp;<br \/>\nFrom here, we can solve the expression in parentheses. <span style=\"font-style:normal; font-size:90%\">\\(1-65,536=-65,535\\)<\/span>, so we can rewrite the equation again.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{2(-65,535)}{1-4}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can simplify the numerator and denominator. <span style=\"font-style:normal; font-size:90%\">\\(2\\times -65,535=-131,070\\)<\/span>, and <span style=\"font-style:normal; font-size:90%\">\\(1-4=-3\\)<\/span>.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{8}=\\frac{-131,070}{-3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we divide to solve. <span style=\"font-style:normal; font-size:90%\">\\(-131,070\\div -3=43,690\\)<\/span>. <\/p>\n<div class=\"examplesentence\">\\(s_{8}=43,690\\)<\/div>\n<p>\n&nbsp;<br \/>\nTherefore, 43,690 people received the YouTube video after 8 rounds of messaging.<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nLet\u2019s try another problem together. Let\u2019s say that Anne\u2019s club is selling cookies online. They sell 4 boxes on day 1. Their goal for each day is to double the amount sold on the previous day. At this rate, how many boxes will they sell on day 10? <\/p>\n<p>In this problem, we need to find the 10th term in the geometric sequence described. <\/p>\n<p>Let\u2019s start by identifying what we know from the problem. Since 4 boxes are sold on Day 1, that\u2019s the first number in our sequence. If their goal is to double, that means that each subsequent term needs to be multiplied by 2. <\/p>\n<p>We can use the first formula we learned to solve this problem. <\/p>\n<div class=\"examplesentence\">\\(a_{n}=a_{1}r^{n-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s replace the variables with their corresponding numbers. <span style=\"font-style:normal; font-size:90%\">\\(n=10\\)<\/span>, since we need to find the 10th term in this sequence. <span style=\"font-style:normal; font-size:90%\">\\(a_{1}=4\\)<\/span>, which is the first number in the sequence. And <span style=\"font-style:normal; font-size:90%\">\\(r=2\\)<\/span> because their goal is to double the previous day\u2019s sales. <\/p>\n<div class=\"examplesentence\">\\(a_{10}=4(2)^{10-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow that the variables are replaced, we can solve the equation. First, we simplify our exponents. Since <span style=\"font-style:normal; font-size:90%\">\\(10-1=9\\)<\/span>, we can rewrite the equation using the exponent 9.<\/p>\n<div class=\"examplesentence\">\\(a_{10}=4(2)^{9}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can rewrite the exponent term. <span style=\"font-style:normal; font-size:90%\">\\(2^{9}=512\\)<\/span>, so we can rewrite the equation using 512. <\/p>\n<div class=\"examplesentence\">\\(a_{10}=4(512)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we multiply to solve. <span style=\"font-style:normal; font-size:90%\">\\(4\\times 512=2,048\\)<\/span>.<\/p>\n<div class=\"examplesentence\">\\(a_{10}=2,048\\)<\/div>\n<p>\n&nbsp;<br \/>\nAt their current rate, Anne\u2019s club will sell 2,048 boxes of cookies on day 10.<\/p>\n<h3><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h3>\n<p>\nI have one last problem for you to try on your own. It combines everything we\u2019ve learned in this video, so it&#8217;s a little more challenging, but I know you can handle it. <\/p>\n<p>Malik opened a bank account in January with $10. Each month, he plans to double the amount deposited in the previous month. Based on this information, how much money will Malik deposit in September? And assuming he doesn\u2019t spend any of the deposited money, how much will Malik have at the end of the year? <\/p>\n<p>This problem asks 2 questions. Let\u2019s start with the first: How much money does Malik plan to deposit in September? See if you can try this part on your own. Pause the video here. When you\u2019re done, press play and check your work. <\/p>\n<p>We can use the sequence formula to identify Malik\u2019s deposit amount in September. Since he makes monthly deposits, this would be his 9th deposit. So:<\/p>\n<div class=\"examplesentence\">\\(a_{n}=a_{1}r^{n-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThat\u2019s the formula we\u2019re gonna use to solve this problem. And <span style=\"font-style:normal; font-size:90%\">\\(n=9\\)<\/span> since September is the 9th month of the year. <span style=\"font-style:normal; font-size:90%\">\\(r=2\\)<\/span> because the sequence doubles each month. And <span style=\"font-style:normal; font-size:90%\">\\(a_{1}=10\\)<\/span> because Malik\u2019s first deposit was 10.<\/p>\n<div class=\"examplesentence\">\\(a_{9}=10(2)^{9-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow that we\u2019ve replaced our variables, we\u2019re ready to solve the equation. First, we\u2019ll simplify our exponents. Since <span style=\"font-style:normal; font-size:90%\">\\(9-1=8\\)<\/span>, we can rewrite the equation using the exponent 8.<\/p>\n<div class=\"examplesentence\">\\(a_{9}=10(2)^{8}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can rewrite the exponent term. <span style=\"font-style:normal; font-size:90%\">\\(2^{8}=256\\)<\/span>, so we can rewrite the equation using 256.<\/p>\n<div class=\"examplesentence\">\\(a_{9}=10(256)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we multiply to solve. <span style=\"font-style:normal; font-size:90%\">\\(10\\times 256=2,560\\)<\/span>.<\/p>\n<div class=\"examplesentence\">\\(a_{9}=2,560\\)<\/div>\n<p>\n&nbsp;<br \/>\nMalik plans to deposit $2,560 in September.<\/p>\n<div class=\"examplesentence\">\\(a_{n}=a_{1}r^{(n-1)}\\)<br \/>\n\\(a_{9}=(10)(2)^{(9-1)}\\)<br \/>\n\\(a_{9}=(10)(2)^{(8)}\\)<br \/>\n\\(a_{9}=(10)(256)\\)<br \/>\n\\(a_{9}=2,560\\)<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s move onto the second question: How much money will Malik have at the end of the year? Pause the video and try to solve this on your own. When you\u2019re done, press play and check your work. <\/p>\n<p>We can use the series formula to find the total amount of money Malik plans to deposit.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{n}=\\frac{a_{1}(1-r^{n})}{1-r}\\)<\/div>\n<p>\n&nbsp;<br \/>\nFirst, we can replace our variables with numbers from the problem. Since we are looking for the total amount deposited after 12 months, <span style=\"font-style:normal; font-size:90%\">\\(n=12\\)<\/span>. We know that <span style=\"font-style:normal; font-size:90%\">\\(a_{1}=10\\)<\/span> because Malik\u2019s first deposit was $10. And we know that <span style=\"font-style:normal; font-size:90%\">\\(r=2\\)<\/span> because our sequence doubles each month.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{12}=\\frac{10(1-2^{12})}{1-2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, we can simplify the exponent. Since <span style=\"font-style:normal; font-size:90%\">\\(2^{12}=4,096\\)<\/span>, we can rewrite the equation.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{12}=\\frac{10(1-4,096)}{1-2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we can solve the expression in parentheses. <span style=\"font-style:normal; font-size:90%\">\\(1-4,096=-4,095\\)<\/span>, so we can rewrite the equation again.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{12}=\\frac{10(-4,095)}{1-2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nOur next step is to simplify the numerator and denominator. <span style=\"font-style:normal; font-size:90%\">\\(10\\times -4,095=-40,950\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(1-2=-1\\)<\/span>.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 110%;\">\\(s_{12}=\\frac{-40,950}{-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we can divide to solve.<\/p>\n<div class=\"examplesentence\">\\(s_{12}=40,950\\)<\/div>\n<p>\n&nbsp;<br \/>\nMalik will deposit $40,950 by the end of the year.<\/p>\n<p>I hope this video on geometric sequences was helpful. Thanks for watching, and happy studying! <\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Geometric_Sequence_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Geometric Sequence Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFind the fifth term in the following geometric sequence:<\/p>\n<div class=\"yellow-math-quote\">5, 30, 180,&#8230;<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\(a_5=6{,}480\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(a_5=1{,}080\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(a_5=4{,}500\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(a_5=5{,}400\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Solving these types of problems can typically be done easiest by first determining the formula for \\(a_n\\), and then plugging in the appropriate \\(n\\)-value.<\/p>\n<p>Since a geometric sequence is defined such that \\(a_n=a_1r^{n-1}\\), determine what \\(a_1\\) and \\(r\\) equal, then plug them in.<\/p>\n<p>The first term of the sequence is 5, so \\(a_1=5\\). The next term is 30, so:<\/p>\n<p style=\"text-align: center\">\\(r=\\dfrac{a_2}{a_1}=\\dfrac{30}{5}=6\\)<\/p>\n<p>Plugging these into the geometric sequence formula, you get \\(a_n=5\\times6^{n-1}\\).<\/p>\n<p>Now, since the problem is asking for the fifth term, plug \\(n=5\\) into this formula and simplify.<\/p>\n<p style=\"text-align: center; line-height: 35px;\">\n\\(a_5=5\\times6^{5-1}\\)<br \/>\n\\(=5\\times6^4\\)<br \/>\n\\(=5\\times1{,}296\\)<br \/>\n\\(=6{,}480\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDetermine the common ratio (\\(r\\)) in the following geometric sequence, and write the formula for \\(a_n\\).<\/p>\n<div class=\"yellow-math-quote\">13, 143, 1573, 17&#8239;303,\u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(r=9\\); \\(a_n=9\\times13^{n-1}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(r=11\\); \\(a_n=13\\times11^{n-1}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(r=13\\); \\(a_n=13\\times11^{n-1}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(r=11\\); \\(a_n=11\\times13^{n-1}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p> The common ratio \\(r\\) can be found by dividing one term by the preceding term.<\/p>\n<p style=\"text-align: center\">\\(r=\\dfrac{a_{n+1}}{a_n}\\)<\/p>\n<p>Usually, the simplest way to determine \\(r\\) is by dividing the first two terms. In this case:<\/p>\n<p style=\"text-align: center\">\\(r=\\dfrac{a_2}{a_1}=\\dfrac{143}{13}=11\\)<\/p>\n<p>Once \\(r\\) is found, you must write the formula for \\(a_n\\). This means you\u2019ll also need to know \\(a_1\\).<\/p>\n<p>By looking at the first term of the sequence above, it is clear that \\(a_1\\) is 13, so the formula for the nth term is \\(a_n=13\\times11^{n-1}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDetermine the sixth term, \\(a_6\\), in the geometric series given by the terms in the geometric sequence below:<\/p>\n<div class=\"yellow-math-quote\">3, 21, 147, 1029, \u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">50,421<\/div><div class=\"PQ\"  id=\"PQ-3-2\">52,620<\/div><div class=\"PQ\"  id=\"PQ-3-3\">64,735<\/div><div class=\"PQ\"  id=\"PQ-3-4\">65,291<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the sixth term in the sequence, it&#8217;s essential to use the formula for the nth term of a geometric sequence:<\/p>\n<p style=\"text-align: center\">\\(a_n=a_1\\times r^{(n-1)}\\)<\/p>\n<p>First, determine the values of \\(a_1\\)and \\(r\\). The first term is 3. The common ratio \\(r\\) can be found by dividing any term of the sequence by the preceding term.<\/p>\n<p>Plugging \\(a_1\\)and \\(r\\) into the formula, and setting \\(n=6\\), we get the following:<\/p>\n<p style=\"text-align:center; line-height: 35px;\">\n\\(a_6=3\\times 7^{(6-1)}\\)<br \/>\n\\(a_6=3\\times 7^5\\)<br \/>\n\\(a_6=3\\times 16{,}807\\)<br \/>\n\\(a_6=50{,}421\\)<\/p>\n<p>Therefore, the sixth term of the geometric sequence is 50,421.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nClaudia is training for a marathon. On her first day of training, she runs for one minute. Each day she will increase her running time by 50%. How much time will she have run cumulatively after seven days? Round to three decimal places.<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">21.818 minutes<\/div><div class=\"PQ\"  id=\"PQ-4-2\">23.541 minutes<\/div><div class=\"PQ\"  id=\"PQ-4-3\">29.940 minutes<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">32.172 minutes<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The word \u201ccumulatively\u201d is a signal that this is a geometric series problem. The formula for the nth term of a geometric series is:<\/p>\n<p style=\"text-align: center\">\\(s_n=\\dfrac{a_1(1-r^n)}{1-r}\\)<\/p>\n<p>First, determine what \\(a_1\\) and \\(r\\) are.<\/p>\n<p>The problem statement says that Claudia runs for one minute on her first day of training, so \\(a_1=1\\). It also says that she increases her running time by 50% each day. Don\u2019t be fooled though\u2014this does not mean that \\(r=0.50\\). If that were the case, then Claudia would only be running half of her previous time each day. Instead, she is increasing her time every day, making \\(r\\) equal to 1.5.<\/p>\n<p>In this case:<\/p>\n<p style=\"text-align: center\">\\(s_n=\\dfrac{1(1-1.5^n)}{1-1.5}\\)<\/p>\n<p>Therefore, Claudia\u2019s cumulative running time after seven days will equal \\(s_7\\).<\/p>\n<p style=\"text-align:center; line-height: 65px;\">\\(s_7=\\dfrac{1(1-1.5^7)}{1-1.5}\\)\\(\\:=\\dfrac{1-1.5^7}{-0.5}\\)\\(\\:=\\dfrac{1-17.086}{-0.5}\\)\\(\\:=\\dfrac{-16.086}{-0.5}\\)\\(\\:=32.172\\text{ minutes}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEvery year on his birthday, Wilbur donates canned food to his local food bank. He started this tradition at age 20 by donating two cans of food. Each year, Wilbur gives triple the amount of the previous year. How many cans of food will Wilbur have donated in total after his 30th birthday?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">84,332<\/div><div class=\"PQ\"  id=\"PQ-5-2\">103,401<\/div><div class=\"PQ\"  id=\"PQ-5-3\">139,577<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">177,146<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Before the formula for the geometric series can be used, first determine what \\(a_1\\) and \\(r\\) are equal to. The first time Wilbur donated food was at age 20, when he gave two cans of food. This means \\(a_1=2\\).<\/p>\n<p>Each year, he triples his donation amount for the year. This indicates that \\(r=3\\).<\/p>\n<p>The geometric series formula is then:<\/p>\n<p style=\"text-align: center\">\\(s_n=\\dfrac{2(1-3^n)}{1-3}\\)<\/p>\n<p>Now determine what \\(n\\)-value should be plugged in. It is tempting to say that \\(n=10\\) after Wilbur\u2019s 30th birthday, but remember that his 20th birthday is included as the first iteration of the geometric series. <\/p>\n<p>Because of this, plug in \\(n=11\\).<\/p>\n<p style=\"text-align: center; line-height: 65px;\">\\(s_{11}=\\dfrac{2(1-3^{11})}{1-3}\\)\\(\\:=\\dfrac{2(1-177{,}147)}{-2}\\)\\(\\:=\\dfrac{2(-177{,}146)}{-2}\\)\\(\\:=177{,}146\\)<\/p>\n<p>After his 30th birthday and 11 years of donations, Wilbur will have given 177,146 cans of food.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #6:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe formula for a geometric sequence is below. What is the 4th term of the geometric sequence?<\/p>\n<div class=\"yellow-math-quote\">\\(a_n= 4 \\cdot 5^{(n-1)}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-6-1\">100<\/div><div class=\"PQ correct_answer\"  id=\"PQ-6-2\">500<\/div><div class=\"PQ\"  id=\"PQ-6-3\">2,500<\/div><div class=\"PQ\"  id=\"PQ-6-4\">8,000<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-6\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-6\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-6-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To find the 4th term of the geometric sequence, we substitute 4 for \\(n\\) in the formula and evaluate to find the value of the term, which is 500, since the first term is 4 and the common ratio is 5, according to the formula.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-6-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-6-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #7:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is the 9th term of the following geometric sequence?<\/p>\n<div class=\"yellow-math-quote\">1, 7, 49, 343, 2401,\u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-7-1\">518,616<\/div><div class=\"PQ\"  id=\"PQ-7-2\">823,543<\/div><div class=\"PQ correct_answer\"  id=\"PQ-7-3\">5,764,801<\/div><div class=\"PQ\"  id=\"PQ-7-4\">40,353,607<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-7\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-7\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-7-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The first term of the geometric sequence is 1 and the common ratio is 7. This tells us that the formula for the geometric sequence is:<\/p>\n<p style=\"text-align: center\">\\(a_n=1 \\cdot 7^{(n-1)}\\)<\/p>\n<p>Therefore:<\/p>\n<p style=\"text-align: center\">\\(a_9=1 \\cdot 7^{(9-1)}=5{,}764{,}801\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-7-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-7-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #8:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is the formula for this geometric sequence?<\/p>\n<div class=\"yellow-math-quote\">3, 12, 48, 192, 768,&#8230;<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-8-1\">\\(a_n=3\\cdot 4^{(n-1)}\\)<\/div><div class=\"PQ\"  id=\"PQ-8-2\">\\(a_n=3\\cdot 9^{(n-1)}\\)<\/div><div class=\"PQ\"  id=\"PQ-8-3\">\\(a_n=4\\cdot 3^{(n-1)}\\)<\/div><div class=\"PQ\"  id=\"PQ-8-4\">\\(a_n=4\\cdot 9^{(n-1)}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-8\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-8\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-8-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>We will start by identifying the first term of the geometric sequence, which is 3. Then we find the common ratio, which is 4 since the terms are increasing by a factor of 4.<\/p>\n<p>Therefore, the formula for this geometric sequence is \\(a_n= 3\\cdot 4^{(n-1)}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-8-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-8-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #9:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is the 5th term of a geometric sequence where the first term is 8 and the common ratio is 3?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-9-1\">648<\/div><div class=\"PQ\"  id=\"PQ-9-2\">1,944<\/div><div class=\"PQ\"  id=\"PQ-9-3\">12,288<\/div><div class=\"PQ\"  id=\"PQ-9-4\">98,304<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-9\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-9\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-9-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since the first term of the geometric sequence is 8 and the common ratio is 3, the formula for this geometric sequence is \\(a_n=8\\cdot 3^{(n-1)}\\).<\/p>\n<p>Therefore, the 5th term can be found using the formula, substituting 5 for \\(n\\) and evaluating:<\/p>\n<p style=\"text-align: center\">\\(a_5=8 \\cdot 3^{(5-1)}=648\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-9-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-9-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #10:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is the 10th term of the following geometric sequence?<\/p>\n<div class=\"yellow-math-quote\">2, 12, 72, 432, 2592,\u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-10-1\">3,072<\/div><div class=\"PQ\"  id=\"PQ-10-2\">6,144<\/div><div class=\"PQ correct_answer\"  id=\"PQ-10-3\">20,155,392<\/div><div class=\"PQ\"  id=\"PQ-10-4\">120,932,352<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-10\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-10\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-10-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Start by identifying the first term of the geometric sequence, which is 2. The common ratio is 6 since the terms are increasing by a factor of 6. Therefore, the formula for this geometric sequence is:<\/p>\n<p style=\"text-align: center\">\\(a_n= 2 \\cdot 6^{(n-1)}\\)<\/p>\n<p>Using this formula, calculate the value of the 10th term of the geometric sequence:<\/p>\n<p style=\"text-align: center\">\\(a_{10}= 2 \\cdot 6^{(10-1)}=20{,}155{,}392\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-10-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-10-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #11:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFind the missing term in the following geometric sequence without using a formula:<\/p>\n<div class=\"yellow-math-quote\">3, 15, 75, 375, ______ , 9375&#8230;<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-11-1\">5<\/div><div class=\"PQ correct_answer\"  id=\"PQ-11-2\">1,875<\/div><div class=\"PQ\"  id=\"PQ-11-3\">1,125<\/div><div class=\"PQ\"  id=\"PQ-11-4\">6,375<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-11\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-11\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-11-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To identify the missing term without a formula, analyze the pattern to find the common ratio.<\/p>\n<p>Start with the first term, 3, and multiply it by what number results in a product of the next term (15), which is 5. This means the common ratio is 5.<\/p>\n<p>Check the rest of the terms in the sequence:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(15 \\times 5=75\\)<br \/>\n\\(75 \\times 5=375\\)<\/p>\n<p>Next, multiply 375 by 5 to find the missing term in the sequence. Since \\(375 \\times 5=1{,}875\\), the missing term in the geometric sequence is 1,875.<\/p>\n<p>Check your work by multiplying 1,875 by 5. If correct, the product should be the subsequent term in the sequence, 9,375. Since \\(1{,}875 \\times 5=9,375\\), Choice B is correct.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-11-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-11-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #12:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nUse the formula \\(a_n=a_1r^{(n-1)}\\) to identify the 8th term in the sequence below: <\/p>\n<div class=\"yellow-math-quote\">1, 6, 36, 216, 1296\u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-12-1\">279,936<\/div><div class=\"PQ\"  id=\"PQ-12-2\">46,656<\/div><div class=\"PQ\"  id=\"PQ-12-3\">48<\/div><div class=\"PQ\"  id=\"PQ-12-4\">6<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-12\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-12\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-12-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, identify the values of the variables used in the equation.<\/p>\n<p>Since \\(n\\) stands for the number in the sequence that needs to be identified, replace \\(n\\) with 8 in the formula. The variable \\(a_1\\) stands for the first term in the sequence, which is 1. The variable \\(r\\) represents the common ratio. In this case, the common ratio is 6.<\/p>\n<p>Each term is multiplied by \\(6\\), resulting in the subsequent term in the sequence. Replace \\(r\\) with \\(6\\) in the formula.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(n=8\\)<br \/>\n\\(a_1=1\\)<br \/>\n\\(r=6\\)<\/p>\n<p>From here, rewrite the formula \\(a_n=a_1r^{(n-1)}\\) and substitute the variables with the numbers they represent:<\/p>\n<p style=\"text-align: center\">\\(a_8=(1)(6)^{(8-1)}\\)<\/p>\n<p>Now that the variables are replaced with their corresponding values, solve the equation using the order of operations.<\/p>\n<p style=\"text-align: center\">\\(a_8=(1)(6)^{(7)}\\)<\/p>\n<p>First, simplify the exponents. Since \\(8-1=7\\), we can rewrite the equation using the exponent 7.<\/p>\n<p style=\"text-align: center\">\\(a_8=(1)(279{,}936)\\)<\/p>\n<p>Next, simplify the exponent 6<sup>7<\/sup>.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(6^7=6 \\times 6 \\times 6 \\times 6 \\times 6 \\times 6 \\times 6\\)\\(\\:= 279{,}936\\)<\/p>\n<p>Rewrite the equation using 279,936.<\/p>\n<p style=\"text-align: center\">\\(a_8=279{,}936\\)<\/p>\n<p>From here, multiply 1 by 279,936 to get the final answer.<\/p>\n<p style=\"text-align: center\">\\(279{,}936 \\times 1=279{,}936\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-12-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-12-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #13:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nUse the formula \\(a_n=a_1r^{(n-1)}\\) to identify the 15th term in the sequence shown: <\/p>\n<div class=\"yellow-math-quote\">7, 21, 63, 189&#8239;567,\u2026<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-13-1\">1,240,029<\/div><div class=\"PQ\"  id=\"PQ-13-2\">3,720,087<\/div><div class=\"PQ\"  id=\"PQ-13-3\">11,160,261<\/div><div class=\"PQ correct_answer\"  id=\"PQ-13-4\">33,480,783<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-13\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-13\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-13-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, identify the values of the variables used in the equation.<\/p>\n<p>Since \\(n\\) stands for the number in the sequence that needs to be identified, replace \\(n\\) with 15 in the formula. The variable \\(a_1\\) stands for the first term in the sequence, which is 7. The variable \\(r\\) represents the common ratio. In this case, the common ratio is 3.<\/p>\n<p>Each term is multiplied by 3, resulting in the subsequent term in the sequence. Replace \\(r\\) with 3 in the formula.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(n=15\\)<br \/>\n\\(a_1=7\\)<br \/>\n\\(r=3\\)<\/p>\n<p>From here, rewrite the formula \\(a_n = a_1r^{(n-1)}\\) and substitute the variables with the numbers they represent:<\/p>\n<p style=\"text-align: center\">\\(a_{15}=(7)(3)^{(15-1)}\\)<\/p>\n<p>Now that the variables are replaced with their corresponding values, solve the equation using the order of operations.<\/p>\n<p style=\"text-align: center\">\\(a_{15}=(7)(3)^{(14)}\\)<\/p>\n<p>First, simplify the exponents. Since \\(15-1=14\\), we can rewrite the equation using the exponent 14.<\/p>\n<p style=\"text-align: center\">\\(a_{15}=(7)(4{,}782{,}969)\\)<\/p>\n<p>Next, simplify the exponent 3<sup>14<\/sup>.<\/p>\n<p class=\"longmath\" style=\"text-align: center\">\\(3^{14}=3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 = 4{,}782{,}969\\)<\/p>\n<p>Rewrite the equation using 4,782,969.<\/p>\n<p style=\"text-align: center\">\\(a_{15}=33{,}480{,}783\\)<\/p>\n<p>From here, multiply 7 by 4,782,969 to get the final answer.<\/p>\n<p style=\"text-align: center\">\\(7 \\times 4{,}782{,}969=33{,}480{,}783\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-13-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-13-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #14:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nKelsey opened a bank account in January and deposited $5. Each month, she plans to double the amount deposited in the previous month. Based on this information, how much money will Kelsey deposit in October?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-14-1\">$10<\/div><div class=\"PQ\"  id=\"PQ-14-2\">$50<\/div><div class=\"PQ\"  id=\"PQ-14-3\">$1,280<\/div><div class=\"PQ correct_answer\"  id=\"PQ-14-4\">$2,560<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-14\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-14\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-14-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To figure out how much money Kelsey plans to deposit, use the sequences formula.<\/p>\n<p>First, identify the values of the variables used in the equation. October will be Kelsey\u2019s tenth deposit, so we are finding the tenth term in the geometric sequence.<\/p>\n<p>Since \\(n\\) stands for the number in the sequence that needs to be identified, replace \\(n\\) with 10 in the formula. The variable \\(a_1\\) stands for the first term in the sequence. Kelsey\u2019s first deposit was in January when she opened her bank account with $5. Therefore, the first term in the sequence is 5.<\/p>\n<p>Each month, Kelsey plans to double the amount of money she deposits. Each term gets doubled, or multiplied by 2, resulting in the subsequent term in the sequence.<\/p>\n<p>Based on this information, the common ratio in this sequence is 2. Therefore, replace \\(r\\) with 2 in the formula.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(n=10\\)<br \/>\n\\(a_1=5\\)<br \/>\n\\(r=2\\)<\/p>\n<p>From here, rewrite the formula \\(a_n = a_1r^{(n-1)}\\) and substitute the variables with the numbers they represent: <\/p>\n<p style=\"text-align: center\">\\(a_{10}=(5)(2)^{(10-1)}\\)<\/p>\n<p>Now that the variables are replaced with their corresponding values, solve the equation using the order of operations.<\/p>\n<p style=\"text-align: center\">\\(a_{10}=(5)(2)^{(9)}\\)<\/p>\n<p>First, simplify the exponents. Since \\(10-1=9\\), we can rewrite the equation using the exponent 9.<\/p>\n<p style=\"text-align: center\">\\(a_{15}=(512)\\)<\/p>\n<p>Next, simplify the exponent 3<sup>9<\/sup>.<\/p>\n<p class=\"longmath\" style=\"text-align: center\">\\(2^2=2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 = 512\\)<\/p>\n<p>Rewrite the equation using 512.<\/p>\n<p style=\"text-align: center\">\\(a_{10}=2{,}560\\)<\/p>\n<p>From here, multiply 5 by 512 to get the final answer.<\/p>\n<p style=\"text-align: center\">\\(5 \\times 512=2{,}560\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-14-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-14-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #15:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nKen is researching the reproduction of a species of bacteria. Based on his research, the bacteria triples in amount every 24 hours. Ken keeps a log of the bacteria count in his laboratory, but the bacteria count on day four of his research is missing. Assuming the bacteria tripled each day as predicted, identify the bacteria count on day four without using the geometric formula:<\/p>\n<div class=\"yellow-math-quote\">8, 24, 72, _____ , 648,&#8230;<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-15-1\">154<\/div><div class=\"PQ\"  id=\"PQ-15-2\">188<\/div><div class=\"PQ correct_answer\"  id=\"PQ-15-3\">216<\/div><div class=\"PQ\"  id=\"PQ-15-4\">459<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-15\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-15\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-15-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To identify the missing term without a formula, analyze the pattern to find the common ratio.<\/p>\n<p>Starting with the first term, 8 multiplied by what number results in a product of the next term, 24? The correct answer is 3, so the common ratio is 3.<\/p>\n<p>Check the rest of the terms in the sequence:<\/p>\n<p style=\"text-align: center\">\\(24 \\times 3=72\\)<\/p>\n<p>Next, multiply 72 by 3 to get the missing term in the sequence. Since \\(72\\times 3=216\\), the missing term in the geometric sequence is 216. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-15-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-15-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/complex-arithmetic\/\">Return to Complex Arithmetic Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Complex Arithmetic Videos<\/p>\n","protected":false},"author":1,"featured_media":100720,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-89032","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-miscellaneous-complex-arithmetic-videos","7":"page_type-video","8":"content_type-practice-questions","9":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/89032","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=89032"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/89032\/revisions"}],"predecessor-version":[{"id":280535,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/89032\/revisions\/280535"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100720"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=89032"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}