{"id":48,"date":"2017-08-23T07:11:27","date_gmt":"2017-08-23T07:11:27","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=48"},"modified":"2017-09-21T19:44:41","modified_gmt":"2017-09-21T19:44:41","slug":"sequences-series","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/sequences-series\/","title":{"rendered":"Sequences &#038; Series"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/compositions-inverse-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-linear-equations-inequalities\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Sequences &amp; Series<\/h1>\n<h4>Objective<\/h4>\n<p>In this section, you will study the definitions of geometric and arithmetic sequences and the formulas for finding\u00a0terms and summations for arithmetic and geometric sequences.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A <strong><em>function\u00a0<\/em><\/strong>is a <strong><em>relation<\/em><\/strong> with an <strong><em>independent\u00a0variable<\/em><\/strong> and a <strong><em>dependent variable<\/em><\/strong>.\u00a0Any value for the independent variable produces exactly one value\u00a0for the dependent variable.<\/li>\n<li>The set of values of the\u00a0independent variable (usually <em><span style=\"text-decoration: none;\">x<\/span><\/em>)\u00a0is the <strong><em>domain<\/em><\/strong>.<\/li>\n<li>The set of values of the dependent variable (usually <em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span>is the <strong><em>range<\/em><\/strong>.<\/li>\n<\/ul>\n<section>\n<h3><strong>What are arithmetic sequences?<\/strong><\/h3>\n<p>A <abbr title=\"an ordered set of mathematical objects\">sequence<\/abbr> is an ordered set of numbers that are related mathematically. It\u00a0is a function in which the domain is the set of <abbr title=\"any of the natural numbers, the negatives of these numbers, and zero \">integers<\/abbr> or the set of <abbr title=\"the number 1 or any number (as 3, 12, 432) obtained by adding 1 to it one or more times\">natural\u00a0numbers<\/abbr>. We use the letter <em><span style=\"text-decoration: none;\">n<\/span><\/em> to stand for\u00a0the independent variable. For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image002.gif\" width=\"82\" height=\"14\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>If we restrict the domain to the set of natural numbers, then\u00a0we can write the equation in functional notation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image004.gif\" width=\"205\" height=\"14\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using our function, we see<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image006.gif\" width=\"60\" height=\"85\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The sequence of numbers is 1, 3, 5, 7, . . . . for\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s6_p2_html_36fc9ce7.gif\" width=\"76\" height=\"21\" name=\"graphics18\" align=\"absmiddle\" border=\"0\" \/>.\u00a0Each value in a sequence is called a <abbr title=\"the individual elements of a polynomial that are added or subtracted\">term<\/abbr>.<br \/>\nIn <abbr title=\"a sequence in which the first term is denoted a1, the second term is a2, etc., through an for the nth term.\">sequence\u00a0notation<\/abbr>, the first term is denoted <em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sub>1<\/sub><\/span>.\u00a0The second term is <em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sub>2<\/sub>,\u00a0<\/span>and so on through <em>a<sub>n<\/sub><\/em> for the nth term. Positive\u00a0integers are represented by <em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0<\/span>We call the nth term the <abbr title=\"the nth term of a sequence\">general\u00a0term<\/abbr>.<\/p>\n<p>If a sequence stops at a given term, it is a <abbr title=\"a sequence that ends with a finite term\">finite\u00a0sequence<\/abbr>. A sequence like the one above, that\u00a0continues indefinitely, is called an <abbr title=\"a sequence that continues indefinitely\">infinite\u00a0sequence<\/abbr>. An infinite sequence is written using an\u00a0ellipsis ( . . . ) to show that it does not end.<\/p>\n<p>Rewriting our sequence in sequence notation gives<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image008.gif\" width=\"66\" height=\"89\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>You can get any term in the sequence by adding two to the\u00a0previous term. A sequence such as this, in which a constant is\u00a0added to the previous term to get the next term, is called an\u00a0<abbr title=\"a sequence of numbers in which a constant is added to the previous number to get the next term.\">arithmetic\u00a0sequence<\/abbr>. The constant is called the <abbr title=\"the constant being added in an arithmetic sequence \">common\u00a0difference<\/abbr>.<\/p>\n<p>If you know the initial term and the common difference, you can\u00a0generate a rule for finding any term by operating on the previous\u00a0term. This rule would be called a <abbr title=\"a rule for finding any term in a sequence by operating on the previous term\">recursive\u00a0formula<\/abbr>. You must know the previous terms to find a\u00a0particular term in a sequence. The recursive formula for the\u00a0sequence we have been working with is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image010.gif\" width=\"76\" height=\"15\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This recursive formula enables us to write an <abbr title=\"formula that gives the value of any term in a sequence without reference to the value of the previous term\">explicit\u00a0formula<\/abbr>. An explicit formula enables us to find the<br \/>\nvalue of any term without knowing the value of the previous term. The explicit formula for our sequence is \u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image012.gif\" width=\"109\" height=\"15\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0where <em><span style=\"text-decoration: none;\">d<\/span><\/em> is the common difference.<\/p>\n<p>Find the 11th term in the sequence.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image014.gif\" width=\"103\" height=\"59\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The sum of the terms in an arithmetic sequence is called an\u00a0<abbr title=\"the sum of the terms in an arithmetic sequence\">arithmetic\u00a0series<\/abbr>. A <abbr title=\"sum of all the terms previous to the selected term of a series\">partial\u00a0sum<\/abbr> of a series adds all the terms prior to the\u00a0selected term. A partial sum can be found even if a sequence is\u00a0infinite. For example,<\/p>\n<p align=\"CENTER\">arithmetic sequence: 1, 3, 5, 7, . . .<\/p>\n<p align=\"CENTER\">arithmetic series: 1 + 3 + 5 + 7 + . . .<\/p>\n<p>Here are a few partial sums for the series\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image030.gif\" width=\"66\" height=\"21\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image032.gif\" width=\"99\" height=\"66\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Use the Greek letter sigma, \u03a3, to write a series in\u00a0series notation. For example, to show that we want to find the 8th\u00a0partial sum of the series we have been working with, use <img loading=\"lazy\" decoding=\"async\" class=\"non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image022.gif\" width=\"65\" height=\"39\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0. The formula below means\u00a0<em><span style=\"text-decoration: none;\">the\u00a0sum from n = 1 to 8 of the terms 2n \u2013 1.<\/span><\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image024.gif\" width=\"288\" height=\"39\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"left\">You can also use the following formula to find the partial sum of an arithmetic sequence:<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/mrcALG1-7a2.gif\" width=\"170\" height=\"36\" \/><\/p>\n<p align=\"left\">Again using the series we have been working with, where <em>n<\/em> = 8, <em>a<\/em><sub>1<\/sub> = (2 \u00d7 1\u00a0\u2212 1) = 1 and <em>d<\/em>, the common difference, equals 2 , we get:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s6_p2_clip_image026.gif\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which is the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image026.gif\" width=\"66\" height=\"39\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>30<\/li>\n<li>18<\/li>\n<li>25<\/li>\n<li>60<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. In expanded notation,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p2_clip_image028.gif\" width=\"220\" height=\"39\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3><strong>What is a geometric sequence?<\/strong><\/h3>\n<p>Recall that in an arithmetic sequence, the difference between\u00a0any two consecutive terms is a constant called the common\u00a0difference. In a geometric\u00a0sequence, or <abbr title=\"a sequence in which there is a common ratio between any two consecutive terms\">geometric\u00a0progression<\/abbr>, there is a common ratio between any two\u00a0consecutive terms. In other words, the division of any term by the\u00a0term just previous to it results in the same number. This ratio is\u00a0called the <abbr title=\"the ratio between any two terms in a geometric sequence \">common\u00a0ratio<\/abbr>.<\/p>\n<p>For example, given the sequence 1, 3, 9, 27, . . .<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image002.gif\" width=\"108\" height=\"230\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>A sequence <em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sub>1<\/sub>,\u00a0<\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sub>2<\/sub>,\u00a0<\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sub>3<\/sub>,\u00a0. . . <\/span><em><span style=\"text-decoration: none;\">a<sub>n\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">is a geometric sequence if for all <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">&gt; 1, <img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image004.gif\" width=\"49\" height=\"34\" name=\"graphics4\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0where <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the common ratio<\/span>.<\/p>\n<p>Using this formula for <em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span>we get a recursive formula for the sequence.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image006.gif\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Note that each time we replaced the new term with the preceding\u00a0term times <em><span style=\"text-decoration: none;\">r.\u00a0<\/span><\/em>We can continue this process until we get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image008.gif\" width=\"140\" height=\"18\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This is the formula to find the <em><span style=\"text-decoration: none;\">n<\/span><\/em>th\u00a0term, also known as the <abbr title=\"the nth term of a geometric sequence \">common\u00a0term<\/abbr>, of a geometric sequence. You can find the\u00a0common term if you know the first term and the common ratio.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which is the sixth term of the geometric sequence 1, 2, 4, 8, .\u00a0. . ?<\/p>\n<ol>\n<li>8<\/li>\n<li>16<\/li>\n<li>32<\/li>\n<li>64<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C. To solve, use the formula\u00a0<img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image010.gif\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image012.gif\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>What is a geometric series?<\/strong><\/h3>\n<p>The summation of the terms in a geometric sequence is a\u00a0geometric\u00a0series. A <abbr title=\"the summation of the terms in a finite geometric sequence\">finite\u00a0geometric series<\/abbr> is the summation of the terms in a\u00a0finite geometric sequence.<\/p>\n<p>For example, if <em><span style=\"text-decoration: none;\">r<\/span><\/em> \u2260 1 is the common ratio, then <img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image014.gif\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Multiply both sides of the equation by <em><span style=\"text-decoration: none;\">r<\/span><\/em>.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image016.gif\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Subtract the two equations.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image018.gif\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Factor the left side.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image020.gif\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image022.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0, and therefore\u00a0<img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image024.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">Finally, <span style=\"font-style: normal;\">replace\u00a0<\/span><\/span><span style=\"font-style: normal;\"><em><span style=\"text-decoration: none;\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s6_p3_html_m7a3cb566.gif\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/span><\/em><span style=\"text-decoration: none;\">with\u00a0<\/span><\/span><span style=\"text-decoration: none;\"><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s6_p3_html_87d261f.gif\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>\u00a0<\/span>in the formula.<\/p>\n<p><img decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image026.gif\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>You can use formula (1) to find the sum of a geometric series\u00a0when you know the first term and the common ratio. When you know\u00a0the first term, the last term, and the common ratio, you can use\u00a0formula (2), which does not require raising the common ratio to\u00a0powers.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the sum of the first 6 terms in the sequence 1, 3, 9, 27, . . . ?<\/p>\n<ol>\n<li>310<\/li>\n<li>156<\/li>\n<li>1365<\/li>\n<li>364<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. We do not know the 6th term, so we\u00a0use formula (1).<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s3_p3_clip_image028.gif\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/compositions-inverse-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-linear-equations-inequalities\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Sequences &amp; Series Objective In this section, you will study the definitions of geometric and arithmetic sequences and the formulas for finding\u00a0terms and summations for arithmetic and geometric sequences. Previously Covered: A function\u00a0is a relation with an independent\u00a0variable and a dependent variable.\u00a0Any value for the independent variable produces exactly [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-48","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/48","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=48"}],"version-history":[{"count":9,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/48\/revisions"}],"predecessor-version":[{"id":789,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/48\/revisions\/789"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=48"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}