{"id":96421,"date":"2022-10-28T07:19:38","date_gmt":"2022-10-28T07:19:38","guid":{"rendered":"https:\/\/gkscientist.com\/?p=96421"},"modified":"2022-10-28T07:19:41","modified_gmt":"2022-10-28T07:19:41","slug":"arithmetico-geometric-series","status":"publish","type":"post","link":"https:\/\/gkscientist.com\/arithmetico-geometric-series\/","title":{"rendered":"Arithmetico-Geometric Series"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_65 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-662c782dbbe2a\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-662c782dbbe2a\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/gkscientist.com\/arithmetico-geometric-series\/#Arithmetico-Geometric_Series\" title=\"Arithmetico-Geometric Series:\">Arithmetico-Geometric Series:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/gkscientist.com\/arithmetico-geometric-series\/#Sum_to_n-terms_of_Arithmetico-Geometric_Series\" title=\"Sum to n-terms of Arithmetico-Geometric Series:\">Sum to n-terms of Arithmetico-Geometric Series:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/gkscientist.com\/arithmetico-geometric-series\/#Sum_of_Infinite_Term_of_Arithmetico-Geometric_Series\" title=\"Sum of Infinite Term of Arithmetico-Geometric Series:\">Sum of Infinite Term of Arithmetico-Geometric Series:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Arithmetico-Geometric_Series\"><\/span>Arithmetico-Geometric Series:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Series of the form <strong><em>a (b) + (a + d) (br)  + (a + 2d) (br<sup>2<\/sup>) + (a + 3d) (br<sup>3<\/sup>) + &#8230;&#8230;&#8230; + (a + (n &#8211; 1) d) (br<sup>n-1<\/sup>) + &#8230;&#8230;&#8230;<\/em><\/strong> is called Arithmetico-Geometric Series (AG Series) which consists of two series i.e.<\/p>\n\n\n\n<ul><li>Arithmetic Series \u2192 a + (a + d) + (a + 2d) + <strong>&#8230;&#8230;&#8230;<\/strong><\/li><li>Geometric Series \u2192 b + br + br<sup>2<\/sup> +  <strong>&#8230;&#8230;&#8230;<\/strong><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Sum_to_n-terms_of_Arithmetico-Geometric_Series\"><\/span>Sum to n-terms of Arithmetico-Geometric Series:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Let S<sub>n<\/sub> = a + (a + d) r + (a + 2d) r<sup>2<\/sup> +  <strong>&#8230;&#8230;&#8230;<\/strong> + (a + (n &#8211; 1) d) r<sup>n-1<\/sup> <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (i)<br>Multiply (i) by r (common ratio):-<br>rS<sub>n<\/sub> = ar + (a + d) r<sup>2<\/sup> + (a + 2d) r<sup>3<\/sup> + <strong>&#8230;&#8230;&#8230;<\/strong> + (a + (n &#8211; 1) d) r<sup>n<\/sup> <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (ii)<br>Subtract (ii) from (i):-<br>S<sub>n<\/sub> &#8211; rS<sub>n<\/sub>= a + dr + dr<sup>2<\/sup> + dr<sup>3<\/sup> + <strong>&#8230;&#8230;&#8230;<\/strong> + dr<sup>n-1<\/sup> &#8211; (a + <img loading=\"lazy\" decoding=\"async\" width=\"40\" height=\"23\" class=\"wp-image-96429\" style=\"width: 40px;\" src=\"https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/for-Arithmetico-Geometric-Series.png?resize=40%2C23&#038;ssl=1\" alt=\"for Arithmetico-Geometric Series\" title=\"\" srcset=\"https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/for-Arithmetico-Geometric-Series.png?w=74&amp;ssl=1 74w, https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/for-Arithmetico-Geometric-Series.png?resize=244%2C142&amp;ssl=1 244w\" sizes=\"(max-width: 40px) 100vw, 40px\" data-recalc-dims=\"1\" \/>d) r<sup>n<\/sup><br>\u21d2 (1 &#8211; r) S<sub>n<\/sub> = a + (dr) (r<sup>n-1<\/sup> &#8211; 1)\/(r &#8211; 1) &#8211; (a + <img decoding=\"async\" style=\"width: 40px;\" src=\"https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/for-Arithmetico-Geometric-Series.png?w=730&#038;ssl=1\" alt=\"for Arithmetico-Geometric Series\" title=\"\" data-recalc-dims=\"1\">d) r<sup>n<\/sup><br>\u21d2 <strong><em>S<sub>n<\/sub> = a\/(1 &#8211; r) + (dr) (1 &#8211; r<sup>n-1<\/sup>)\/(1 &#8211; r)<sup>2<\/sup> &#8211; (a + <img decoding=\"async\" style=\"width: 40px;\" src=\"https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/for-Arithmetico-Geometric-Series.png?w=730&#038;ssl=1\" alt=\"for Arithmetico-Geometric Series\" title=\"\" data-recalc-dims=\"1\">d) r<sup>n<\/sup>\/(1 &#8211; r)<\/em><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Sum_of_Infinite_Term_of_Arithmetico-Geometric_Series\"><\/span>Sum of Infinite Term of Arithmetico-Geometric Series:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Let S<sub>\u221e<\/sub> = a + (a + d) r + (a + 2d) r<sup>2<\/sup> + &#8230;&#8230;.. \u221e <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (i)<br>Multiplying (i) by r:-<br>rS<sub>\u221e<\/sub> = ar + (a + d) r<sup>2<\/sup> + (a + 2d) r<sup>3<\/sup> + &#8230;&#8230;.. \u221e <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (ii)<br>Subtract (ii) from (i):-<br>S<sub>\u221e<\/sub> &#8211; rS<sub>\u221e<\/sub>= a + dr + dr<sup>2<\/sup> + &#8230;&#8230;.. \u221e<br>\u21d2 (1 &#8211; r) S<sub>\u221e<\/sub>= a + dr\/(1 &#8211; r)<br>\u21d2 <strong><em>S<sub>\u221e<\/sub>= a\/(1 &#8211; r) + dr\/(1 &#8211; r)<sup>2<\/sup><\/em><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong><em>Example- <mark style=\"background-color:rgba(0, 0, 0, 0);color:#2196f3\" class=\"has-inline-color\">Show that 1 + 2x + 3x<sup>2<\/sup> + &#8230;. to n terms (x &lt; 1) is (1 &#8211; x<sup>n<\/sup>)\/(1 &#8211; x)<sup>2<\/sup> &#8211; nx<sup>n<\/sup>\/(1 &#8211; x).<\/mark><\/em><\/strong> <strong><em><mark style=\"background-color:rgba(0, 0, 0, 0);color:#2196f3\" class=\"has-inline-color\">Find also the sum when n \u2192 \u221e.<\/mark><\/em><\/strong><br><br><strong><em>Solution-<\/em><\/strong> The given series is, <br>1 + 2x + 3x<sup>2<\/sup> + &#8230;&#8230; to n terms<br><br>The corresponding A.P. is 1, 2, 3, &#8230;&#8230;&#8230;<br>It&#8217;s n<sup>th<\/sup> term is 1 + (n &#8211; 1) 1<br>= 1 + n &#8211; 1<br>=n<br><br>The corresponding G.P. is 1, x, x<sup>2<\/sup>, &#8230;&#8230;&#8230;<br>Its n<sup>th <\/sup>term is 1. x<sup>n-1<\/sup><br>= x<sup>n-1<\/sup><br><br>\u2234 the n<sup>th<\/sup> term of the given series = (n<sup>th<\/sup> term of A.P.) (n<sup>th<\/sup> term of G.P.) = nx<sup>n-1<\/sup><br><br>Let S<sub>n<\/sub> = 1 + 2x + 3x<sup>2<\/sup> + &#8230;&#8230; + (n &#8211; 1) x<sup>n-2<\/sup> + nx<sup>n-1<\/sup> <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (i)<br>Multiply (i) by x:-<br>xS<sub>n<\/sub> = x + 2x<sup>2<\/sup> + 3x<sup>3<\/sup> + &#8230;&#8230; + (n &#8211; 1) x<sup>n-1<\/sup> + nx<sup>n<\/sup> <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (ii)<br>Subtract (ii) from (i):-<br>S<sub>n<\/sub> &#8211; xS<sub>n<\/sub> = 1 + x + x<sup>2<\/sup> + x<sup>3<\/sup> + &#8230;&#8230; + x<sup>n-1<\/sup> &#8211; nx<sup>n<\/sup><br>\u21d2 (1 &#8211; x) S<sub>n<\/sub> = Sum of n term of G.P. with a = 1, r = x &#8211; nx<sup>n<\/sup><br>\u21d2 (1 &#8211; x) S<sub>n<\/sub> = [1(1 &#8211; x<sup>n<\/sup>)\/(1 &#8211; x)] &#8211; nx<sup>n<\/sup><br>\u21d2 S<sub>n<\/sub> = [(1 &#8211; x<sup>n<\/sup>)\/(1 &#8211; x)<sup>2<\/sup>] &#8211; [nx<sup>n<\/sup>\/(1 &#8211; x)] <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (A)<br>As n \u2192 \u221e and x &lt; 1<br>\u2234 x<sup>n<\/sup> \u2192 0<br>\u2234 Equation (A) becomes:-<br>S<sub>\u221e<\/sub> = [(1 &#8211; 0)\/(1 &#8211; x)<sup>2<\/sup>] &#8211; [n (0)\/(1 &#8211; x)]<br>S<sub>\u221e<\/sub> = 1\/(1 &#8211; x)<sup>2<\/sup><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong><em>Example-<\/em><\/strong> <strong><em><mark style=\"background-color:rgba(0, 0, 0, 0);color:#2196f3\" class=\"has-inline-color\">Sum to n terms of the series 1 + 2\/3 + 3\/3<sup>2<\/sup> + 4\/3<sup>3<\/sup> + &#8230;&#8230;&#8230;..<\/mark><\/em><\/strong><br><br><strong><em>Solution-<\/em><\/strong>The given series is,<br>1 + 2\/3 + 3\/3<sup>2<\/sup> + 4\/3<sup>3<\/sup> + &#8230;&#8230;&#8230;..<br><br>Let T<sub>n<\/sub> be its n<sup>th<\/sup> term.<br>T<sub>n<\/sub> = (n<sup>th<\/sup> term of A.P. 1, 2, 3, &#8230;&#8230;&#8230;) (n<sup>th<\/sup> term of G.P. 1, 1\/3, 1\/3<sup>2<\/sup>, &#8230;&#8230;&#8230;..)<br>\u21d2 T<sub>n<\/sub> = [1 + (n &#8211; 1)1] [1 (1\/3)<sup>n-1<\/sup>]<br>\u21d2 T<sub>n<\/sub> = n (1\/3)<sup>n-1<\/sup><br><br>Let S<sub>n<\/sub> = 1 + 2\/3 + 3\/3<sup>2<\/sup> + 4\/3<sup>3<\/sup> + &#8230;&#8230;&#8230;.. + (n &#8211; 1) 1\/3<sup>n-2<\/sup> + n (1\/3<sup>n-1<\/sup>) <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (i)<br>Multiply (i) by 1\/3<br>(1\/3) S<sub>n<\/sub> = 1\/3 + 2\/3<sup>2<\/sup> + 3\/3<sup>3<\/sup> + &#8230;&#8230;&#8230;.. + (n &#8211; 1)\/3<sup>n-1<\/sup> + n\/3<sup>n<\/sup> <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (ii)<br><br>Subtract (ii) from (i):-<br>S<sub>n<\/sub> &#8211; (1\/3) S<sub>n<\/sub> = 1 + 1\/3 + 1\/3<sup>2<\/sup> + 1\/3<sup>3<\/sup> + &#8230;&#8230;&#8230;.. + 1\/3<sup>n-1<\/sup> &#8211; n\/3<sup>n<\/sup>        <br>\u21d2 (1 &#8211; 1\/3) S<sub>n<\/sub> = (Sum of n term of G.P. with a = 1, r = 1\/3) &#8211; n\/3<sup>n<\/sup><br>\u21d2 (2\/3) S<sub>n<\/sub> = 1[1 &#8211; (1\/3<sup>n<\/sup>)]\/[1 &#8211; (1\/3)] &#8211; n\/3<sup>n<\/sup><br>\u21d2 (2\/3) S<sub>n<\/sub> = (3<sup>n<\/sup> &#8211; 1)\/3<sup>n<\/sup>(2\/3) &#8211;  n\/3<sup>n<\/sup><br>\u21d2 S<sub>n<\/sub> = (3<sup>n<\/sup> &#8211; 1)\/3<sup>n<\/sup>(2\/3)<sup>2<\/sup> &#8211; n\/(3<sup>n<\/sup>) (2\/3)<br>\u21d2 S<sub>n<\/sub> = 1\/(3<sup>n<\/sup>) (2\/3) [(3<sup>n<\/sup> &#8211; 1)\/(2\/3) &#8211; n]<br>\u21d2 S<sub>n<\/sub> = 1\/(2) (3<sup>n-1<\/sup>) [(3\/2) (3<sup>n<\/sup> &#8211; 1) &#8211; n]<br>\u21d2 S<sub>n<\/sub> = 1\/(2) (3<sup>n-1<\/sup>) [(3<sup>n+1<\/sup> &#8211; 3 &#8211; 2n)\/2]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong><em>Example-<\/em><\/strong> <strong><em><mark style=\"background-color:rgba(0, 0, 0, 0);color:#2196f3\" class=\"has-inline-color\">If the sum to infinity of the series 3 + 5r + 7r<sup>2<\/sup> + &#8230;.. is 44\/9, find r.<\/mark><\/em><\/strong><br><br><strong><em>Solution-<\/em><\/strong> Let S<sub>\u221e<\/sub> = 3 + 5r + 7r<sup>2<\/sup> + &#8230;.. \u221e <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (i)<br>Multiply (i) by r:-<br>rS<sub>\u221e<\/sub> = 3r + 5r<sup>2<\/sup> + 7r<sup>3<\/sup> + &#8230;..  <strong>&#8230;&#8230;&#8230;<\/strong>&#8230;. (ii)<br><br>Subtract (ii) from (i):-<br>S<sub>\u221e<\/sub> &#8211; rS<sub>\u221e<\/sub> = 3 + 2r + 2r<sup>2<\/sup> + 2r<sup>3<\/sup> + &#8230;.. \u221e<br>\u21d2 (1 &#8211; r) S<sub>\u221e<\/sub> = 3 + 2r\/(1 &#8211; r)<br>\u21d2 S<sub>\u221e<\/sub> = 3\/(1 &#8211; r) + 2r\/(1 &#8211; r)<sup>2<\/sup><br>\u21d2 S<sub>\u221e<\/sub> = [3 (1 &#8211; r) + 2r]\/(1 &#8211; r)<sup>2<\/sup><br>\u21d2 S<sub>\u221e<\/sub> = (3 &#8211; r)\/(1 &#8211; r)<sup>2<\/sup><br><br>But S<sub>\u221e<\/sub> = 44\/9<br><br>\u2234 (3 &#8211; r)\/(1 &#8211; r)<sup>2<\/sup> = 44\/9<br>\u21d2 27 &#8211; 9r = 44 (1 &#8211; r)<sup>2<\/sup><br>\u21d2 27 &#8211; 9r = 44 (1 + r<sup>2<\/sup> &#8211; 2r)<br>\u21d2 27 &#8211; 9r = 44 + 44r<sup>2<\/sup> &#8211; 88r<br>\u21d2 -44r<sup>2<\/sup> + 88r &#8211; 9r = 44 &#8211; 27<br>\u21d2 -44r<sup>2<\/sup> + 79r = 17<br>\u21d2 79r = 44r<sup>2<\/sup> + 17<br>\u21d2 44r<sup>2<\/sup> &#8211; 79r + 17 = 0<br>\u21d2 44r<sup>2<\/sup> &#8211; 68r &#8211; 11r + 17 = 0<br>\u21d2 4r (11r &#8211; 17) &#8211; 1 (11r &#8211; 17) = 0<br>\u21d2 (4r &#8211; 1) (11r &#8211; 17) = 0<br>Either 4r &#8211; 1 = 0 or 11r &#8211; 17 = 0<br>r = 1\/4 or r = 17\/11<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator aligncenter has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><a href=\"https:\/\/gkscientist.com\/arithmetic-sequence-formulas\/\" target=\"_blank\" rel=\"noreferrer noopener\">Arithmetic Sequence Formulas<\/a><br><a href=\"https:\/\/gkscientist.com\/arithmetic-mean-in-sequence-and-series\/\" target=\"_blank\" rel=\"noreferrer noopener\">Arithmetic Mean in Sequence and Series<\/a><br><a href=\"https:\/\/gkscientist.com\/geometric-sequence-formulas\/\" target=\"_blank\" rel=\"noreferrer noopener\">Geometric Sequence Formulas<\/a><br><a href=\"https:\/\/gkscientist.com\/geometric-mean-in-sequence-and-series\/\" target=\"_blank\" rel=\"noreferrer noopener\">Geometric Mean in Sequence and Series<\/a><br><a href=\"https:\/\/en.wikipedia.org\/wiki\/Arithmetico-geometric_sequence\" target=\"_blank\" rel=\"noreferrer noopener\">Arithmetico-geometric sequence<\/a>&#8211; Wikipedia<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Arithmetico-Geometric Series: Series of the form a (b) + (a + d) (br) + (a + 2d) (br2) + (a + 3d) (br3) + &#8230;&#8230;&#8230; + (a + (n &#8211; 1) d) (brn-1) + &#8230;&#8230;&#8230; is called Arithmetico-Geometric Series (AG Series) which consists of two series i.e. Arithmetic Series \u2192 a + (a + d) + (a + 2d) + &#8230;&#8230;&#8230; Geometric Series \u2192 b + br + br2 + &#8230;&#8230;&#8230; Sum to n-terms of Arithmetico-Geometric Series: Let Sn = a + (a + d) r + (a + 2d) r2 + &#8230;&#8230;&#8230; + (a + (n &#8211; 1) d) <\/p>\n","protected":false},"author":1,"featured_media":96423,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[5642],"tags":[7407,7408],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/i0.wp.com\/gkscientist.com\/wp-content\/uploads\/2022\/10\/Arithmetico-Geometric-Series.jpg?fit=500%2C250&ssl=1","_links":{"self":[{"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/posts\/96421"}],"collection":[{"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/comments?post=96421"}],"version-history":[{"count":0,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/posts\/96421\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/media\/96423"}],"wp:attachment":[{"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/media?parent=96421"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/categories?post=96421"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gkscientist.com\/wp-json\/wp\/v2\/tags?post=96421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}