{"id":9055,"date":"2025-11-30T13:53:03","date_gmt":"2025-11-30T08:23:03","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9055"},"modified":"2025-11-30T13:53:12","modified_gmt":"2025-11-30T08:23:12","slug":"sequence-and-series-formula-for-jee-main-advanced","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/sequence-and-series-formula-for-jee-main-advanced\/","title":{"rendered":"Sequence and Series Formula for JEE Main\/Advanced"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Sequence<\/h2>\n\n\n\n<p>It means an arrangement of number in definite order according to some rule<\/p>\n\n\n\n<p><strong>Important Points<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The various numbers occurring in a sequence are called its terms. The number of terms is called the length of the series<\/li>\n\n\n\n<li>Terms of a sequence are denoted generally by $a_1 , a_2, a_3, \u2026.., a_n$ etc., The subscripts denote the position of the term. We can choose any other letter to denote it<\/li>\n\n\n\n<li>The nth term is the number at the nth position of the sequence and is denoted by a<sub>n<\/sub><\/li>\n\n\n\n<li>The nth term is also called the general term of the sequence and given by $t_n = S_n &#8211; S_{n-1}$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Arithmetic Progression<\/h2>\n\n\n\n<p>An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant<\/p>\n\n\n\n<p>a, a+ d, a+2d. a+ 3d<\/p>\n\n\n\n<p><strong>Important formula<\/strong><\/p>\n\n\n\n<p>1. How to Prove  if the sequence is  AP given nth term<\/p>\n\n\n\n<p>if $T_n &#8211; T_{n-1}$ is independent of n and is constant<\/p>\n\n\n\n<p>Generally if $T_n= An + B$ , then it will be in AP<\/p>\n\n\n\n<p>2. nth Term of A.P from beginning<\/p>\n\n\n\n<p>$T_n=a + (n-1)d  = l$<\/p>\n\n\n\n<p>3. nth Term of A.P from end<\/p>\n\n\n\n<p>$T_n^{&#8216;}=l &#8211; (n-1)d =a$<\/p>\n\n\n\n<p>4. $T_n + T_n^{&#8216;} = a + l$<\/p>\n\n\n\n<p>This also proves that  the sum of the terms equidistant from beginning and end are equal  and is same as a +l<\/p>\n\n\n\n<p>i.e if an A.P has 5 terms $a_1, a_2,a_3,a_4,a_5$,then<\/p>\n\n\n\n<p>$a_1 + a_5= a_2 + a_4 = a_3 + a_3$<\/p>\n\n\n\n<p>5. if a, b  and c are in A.P, then<\/p>\n\n\n\n<p>$2b= a+c$<\/p>\n\n\n\n<p>6. if a  and l are first and last term of the A.P of n items, then<\/p>\n\n\n\n<p>$d= \\frac {l-a}{n-1}$<\/p>\n\n\n\n<p>6. Sum of the n terms of the A.P<\/p>\n\n\n\n<p>$S_n=\\frac {n}{2} [a +l]$<\/p>\n\n\n\n<p>$S_n=\\frac {n}{2} [2a + (n-1)d]$<\/p>\n\n\n\n<p>$S_n=\\frac {n}{2} [T_n+ T_n^{&#8216;}]$<\/p>\n\n\n\n<p>7. Some General Formulas Based on Above<\/p>\n\n\n\n<p>$1 + 2 + 3 + 4&#8230;+ n= \\frac {n(n+1)}{2}$<\/p>\n\n\n\n<p>$1 + 3 + 5..+ 2n-1= n^2$<\/p>\n\n\n\n<p>8. Some More Formula&#8217;s<\/p>\n\n\n\n<p>$d= T_n &#8211; T_{n-1}$<br>$T_n= S_n-S_{n-1}$<br>$d= S_n &#8211; 2S_{n-1} + S_{n-2}$<\/p>\n\n\n\n<p>9.How to Prove  if the sequence is  AP given $S_n$<\/p>\n\n\n\n<p>if $S_n$ is of the form, $S_n= An^2 + Bn$  then it will be in AP<\/p>\n\n\n\n<p>10. If we have to choose three members of A.P<\/p>\n\n\n\n<p>a-d, a a +d<\/p>\n\n\n\n<p>11. If we have to choose four members of A.P<\/p>\n\n\n\n<p>a- 3d, a-d, a+ d, a+ 3d<\/p>\n\n\n\n<p>12. If we have to choose five members of A.P<\/p>\n\n\n\n<p>a- 2d, a-d, a, a+ d, a+ 2d<\/p>\n\n\n\n<p>13. Arithmetic Mean is given by<\/p>\n\n\n\n<p>$AM= \\frac {a+b}{2}$<\/p>\n\n\n\n<p>14. if $a_1$, $a_2$,$a_3$&#8230; are in AP, then<\/p>\n\n\n\n<p>$a_1 + k$, $a_2 + k$, $a_3 + k$, .. are also in AP<\/p>\n\n\n\n<p>$a_1 &#8211; k$, $a_2 &#8211; k$, $a_3 &#8211; k$, .. are also in AP<\/p>\n\n\n\n<p>$ka_1$, $ka_2$, $ka_3$, .. are also in AP.<\/p>\n\n\n\n<p>$\\frac {a_1}{k}$, $\\frac {a_2}{k}$, $\\frac {a_3}{k}$, .. are also in AP.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Geometric Progressions<\/h3>\n\n\n\n<p>A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout.<\/p>\n\n\n\n<p>$a, ar, ar^2 ,ar^3$<\/p>\n\n\n\n<p>Common difference =r<br>$a  \\ne 0$ and $ r \\ne 0$<\/p>\n\n\n\n<p><strong>Important formula<\/strong><\/p>\n\n\n\n<p>1.nth term of GP from Beginning<\/p>\n\n\n\n<p>$T_n=ar^{n-1}=l$<\/p>\n\n\n\n<p>2. nth term of GP from last<\/p>\n\n\n\n<p>$T_n^{&#8216;}=\\frac {l}{r^{n-1}}=a$<\/p>\n\n\n\n<p>3. $T_n \\times T_n^{&#8216;} = a \\times l$<\/p>\n\n\n\n<p>It also proves that Product of kth term from beginning and last  is independent and is equal to $a \\times l$<\/p>\n\n\n\n<p>4. if a, b and c are in G.P<br>$b^2 = ac$<\/p>\n\n\n\n<p>5. If a  and l are first and last  term of the G.P<\/p>\n\n\n\n<p>$r= (\\frac {l}{a})^{1\/n-1}$<\/p>\n\n\n\n<p>6. Sum of G.P<\/p>\n\n\n\n<p>$S_n =a \\frac {r^n -1}{r-1}$  if r &gt;1<\/p>\n\n\n\n<p>$S_n =a \\frac {r^n -1}{r-1}$  if r &gt;1<br>$S_n = na$  if r =1<\/p>\n\n\n\n<p>7. if the G.P is infinite<\/p>\n\n\n\n<p>if |r| &lt; 1<\/p>\n\n\n\n<p>then $S_n = \\frac {a}{1-r}$<\/p>\n\n\n\n<p>8.  if $a_1, a_2,a_3 ..a_n$ are in G.P<\/p>\n\n\n\n<p>then <\/p>\n\n\n\n<p>$\\frac {1}{a_1} , \\frac {1}{a_2}, \\frac {1}{a_3} &#8230;. , \\frac {1}{a_n}$ are in G.P with common difference  $\\frac {1}{r}$<\/p>\n\n\n\n<p>$a_1^n , a_2^n, a_3^n  ..$ are in G.P with common difference $r^n$<\/p>\n\n\n\n<p>Also<\/p>\n\n\n\n<p>$a_1a_n= a_2a_{n-1} = a_3a_{n-2}=..$<\/p>\n\n\n\n<p>9. if the three number of G.P  are to be taken<\/p>\n\n\n\n<p>$\\frac {a}{r}, a , ar$<\/p>\n\n\n\n<p>10.   if the five number of G.P  are to be taken<\/p>\n\n\n\n<p>$\\frac {a}{r^2},\\frac {a}{r}, a , ar,ar^2$<\/p>\n\n\n\n<p>11. If $a_1,a_2 ,a_3$ are in G.P<\/p>\n\n\n\n<p>$log a_1 , log a_2, log a_3$ are in A.P<\/p>\n\n\n\n<p>12. Geometric Mean is given by<\/p>\n\n\n\n<p>$GM= \\sqrt {ab}$<\/p>\n\n\n\n<p>13.if $a_1$, $a_2$,$a_3$&#8230; are in GP, then<\/p>\n\n\n\n<p>$ka_1$, $ka_2$, $ka_3$, .. are also in GP.<\/p>\n\n\n\n<p>$\\frac {a_1}{k}$, $\\frac {a_2}{k}$, $\\frac {a_3}{k}$, .. are also in GP.<\/p>\n\n\n\n<p>$\\frac {1}{a_1}$, $\\frac {1}{a_2}$, $\\frac {1}{a_3}$, .. are also in GP.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Formula based on Geometric Progression<\/h2>\n\n\n\n<p>(a) Recurring Decimal<\/p>\n\n\n\n<p>R=.PQQQ&#8230;.<br>let p be number of digits in P and q be the number of digits in Q<br>$10^p R =P.QQQQ&#8230;$<br>$10^{p+q} R = PQQQQQ&#8230;$<\/p>\n\n\n\n<p>Subtracting above two<\/p>\n\n\n\n<p>$R= \\frac {PQ -P}{10^{p+q} &#8211; 10^p}$<\/p>\n\n\n\n<p>Example <br>R=.32585858..<br>Then<br>$R= \\frac {3258 -32}{10^4 -10^2} = \\frac {3226}{9900}= \\frac {1613}{4950}$<\/p>\n\n\n\n<p>If R=.QQQQ<br>then<br>$R= \\frac {Q}{10^q -1}$<\/p>\n\n\n\n<p>(b)  $S_n= a + aa + aaa + &#8230;.$  where $a \\n N , 1 \\leq a \\leq 9$<br>example<br>$S_1 = 1 + 11 + 111 + 1111 +&#8230;.$<br>$S_2 = 2 + 22 + 222..$<br>then<br>$S_n= \\frac {a}{9} [ \\frac {10}{9} (10^n -1) -n]$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Inequalities <\/h2>\n\n\n\n<p>A.M &gt; G.M<\/p>\n\n\n\n<p>$a+ b \\geq 2 \\sqrt {ab}$<\/p>\n\n\n\n<p>$\\frac {a_1 + a_2 + a_3 + &#8230;a_n}{n}  \\geq (a_1a_2a_3..a_n)^{1\/n}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Arithmetic Geometric Series<\/h2>\n\n\n\n<p>Let $S =a + (a+d)r + (a+2d)r^2 +&#8230;.. + [a + (n&#8211;1)d]r^{n-1}$<\/p>\n\n\n\n<p>Here we have both the AP and G.P in the terms. Lets see how to to calculate the sum<\/p>\n\n\n\n<p>$rS= a r+ (a+d)r^2 + (a+2d)r^3 +&#8230;.. + [a + (n-1)d]r^{n}$<\/p>\n\n\n\n<p>Subtracting we get<\/p>\n\n\n\n<p>$(1-r)S= a + (dr  + dr^2 + ..dr^{n-1}) -[a + (n-1)d]r^{n}$<br>$1-r)S= a +\\frac {dr(1-r^{n-1})}{1-r} -[a + (n-1)d]r^{n}$<\/p>\n\n\n\n<p>or<\/p>\n\n\n\n<p>$S= \\frac {a}{1-r} +\\frac {dr(1-r^{n-1})}{(1-r)^2} -\\frac {[a + (n-1)d]r^{n}}{1-r}$<\/p>\n\n\n\n<p>when |r| &lt; 1. $n -&gt; \\infty$, $r^n=0$, hence<\/p>\n\n\n\n<p>$S=frac {a}{1-r} +\\frac {dr}{(1-r)^2} <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Important Formula on Sigma<\/h2>\n\n\n\n<p>$\\sum_{k=1}^{n} T_k= T_1 + T_2 + T_3 +\u2026+T_n$<\/p>\n\n\n\n<p>$\\sum_{k=1}^{n} (T_k \\pm T^{&#8216;}_k=\\sum_{k=1}^{n} T_k \\pm \\sum_{k=1}^{n} T_k^{&#8216;}$<\/p>\n\n\n\n<p>$\\sum_{r=1}^{n} f(r+1) &#8211; f(r)= f(n+1) &#8211; f(1)$<\/p>\n\n\n\n<p>$\\sum_{r=1}^{n} f(r+2) &#8211; f(r)= f(n+2) + f(n+1) &#8211; f(2) -f(1)$<\/p>\n\n\n\n<p>$\\sum_{n=1}^{n} n = \\frac {n(n+1)}{2}$<\/p>\n\n\n\n<p>$\\sum_{n=1}^{n} n^2 = \\frac {n(n+1)(2n+1)}{6}$<\/p>\n\n\n\n<p>$\\sum_{n=1}^{n} n^3 = [\\frac {n(n+1)}{2}]^2$<\/p>\n\n\n\n<p>$\\sum_{n=1}^{n} n^4 = \\frac {n(n+1)(2n+1)(3n^2 + 3n -1)}{30}$<\/p>\n\n\n\n<p>$\\sum_{n=1}^{n} a^n =\\frac {a(a^n -1)}{a-1}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Rules for finding the sum of Series<\/h3>\n\n\n\n<p>a. Write the nth term T<sub>n<\/sub>of the series<\/p>\n\n\n\n<p>b. Write the T<sub>n<\/sub> in the <a href=\"https:\/\/physicscatalyst.com\/Class9\/polynomial.php\">polynomial<\/a> form of n<\/p>\n\n\n\n<p>$T_n= a n^3 + bn^2 + cn +d$<\/p>\n\n\n\n<p>c. The sum of series can be written as<\/p>\n\n\n\n<p>$ \\sum S = a \\sum {n^3} + b \\sum {n^2} + c \\sum n + nd$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Method of Difference<\/h3>\n\n\n\n<p>Many times , nth term of the series can be determined. For example<\/p>\n\n\n\n<p>5 + 11 + 19 + 29 + 41\u2026\u2026<\/p>\n\n\n\n<p>If the series is such that difference between successive terms are either in A.P or G.P, then we can the nth term using method of difference<\/p>\n\n\n\n<p>$S_n = 5 + 11 + 19 + 29 + \u2026 + a_{n-1} + a_n$<\/p>\n\n\n\n<p>or $S_n= 5 + 11 + 19 + \u2026 + + a_{n-1} + a_n$<\/p>\n\n\n\n<p>On subtraction, we get<\/p>\n\n\n\n<p>$0 = 5 + [6 + 8 + 10 + 12 + \u2026(n \u2013 1) terms] \u2013 a_n$<\/p>\n\n\n\n<p>Here 6,8,10 is in A.P,So<\/p>\n\n\n\n<p>$a_n = 5 + \\frac {n-1}{2} [12 + (n-1)2]$<\/p>\n\n\n\n<p>or $ a_n= n^2 + 3n + 1$<\/p>\n\n\n\n<p>Now it is easy to find the Sum of the series<\/p>\n\n\n\n<p>$S_n = \\sum_{k=1}^{n} {k^2 +3k +1}$<\/p>\n\n\n\n<p>$=\\frac {n(n+2)(n+4)}{3}$<\/p>\n\n\n\n<p><strong>Short Cut Formula<\/strong><\/p>\n\n\n\n<p>If the First difference is in AP or GP<\/p>\n\n\n\n<p>for AP<\/p>\n\n\n\n<p>$T_n= a(n-1)(n-2) + b(n-1) + c$<\/p>\n\n\n\n<p>we can find a,b,c using terms 1,2, 3<\/p>\n\n\n\n<p>for GP<\/p>\n\n\n\n<p>$T_n= ar^{n-1} + bn + c$<\/p>\n\n\n\n<p>If the difference of the differences are in AP or GP<\/p>\n\n\n\n<p>for AP<\/p>\n\n\n\n<p>$T_n= a(n-1)(n-2)(n-3) + b(n-1)(n-2) + c(n-1) + d$<\/p>\n\n\n\n<p>we can find a,b,c ,d  using terms 1,2, 3,4<\/p>\n\n\n\n<p>for GP<\/p>\n\n\n\n<p>$T_n= ar^{n-1} + bn^2 + cn +d$<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sequence It means an arrangement of number in definite order according to some rule Important Points Arithmetic Progression An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant a, a+ d, a+2d. a+ 3d Important formula 1. How to Prove if the sequence is AP [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[24],"tags":[],"class_list":["post-9055","post","type-post","status-publish","format-standard","hentry","category-general"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Sequence and Series Formula for JEE Main\/Advanced - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/sequence-and-series-formula-for-jee-main-advanced\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sequence and Series Formula for JEE Main\/Advanced - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"Sequence It means an arrangement of number in definite order according to some rule Important Points Arithmetic Progression An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant a, a+ d, a+2d. a+ 3d Important formula 1. 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