{"id":28134,"date":"2022-01-13T17:59:36","date_gmt":"2022-01-13T12:29:36","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=28134"},"modified":"2022-01-13T17:59:36","modified_gmt":"2022-01-13T12:29:36","slug":"sequences-and-series-class-11-notes-maths-chapter-9","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/","title":{"rendered":"Sequences and Series Class 11 Notes Maths Chapter 9"},"content":{"rendered":"<p><strong>Sequence<\/strong><br \/>\nA succession of numbers arranged in a definite order according to a given certain rule is called sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence.<\/p>\n<p><strong>Series<\/strong><br \/>\nIf a<sub>1<\/sub>, a<sub>2<\/sub>, a<sub>3<\/sub>,\u2026\u2026 a<sub>n<\/sub> is a sequence, then the expression a<sub>1<\/sub> + a<sub>2<\/sub> + a<sub>3<\/sub> + a<sub>4<\/sub> + \u2026 + a<sub>n<\/sub> is called series.<\/p>\n<p><strong>Progression<\/strong><br \/>\nA sequence whose terms follow certain patterns are more often called progression.<\/p>\n<p><strong>Arithmetic Progression (AP)<\/strong><br \/>\nA sequence in which the difference of two consecutive terms is constant, is called Arithmetic progression (AP).<\/p>\n<p><strong>Properties of Arithmetic Progression (AP)<\/strong><br \/>\nIf a sequence is an A.P. then its nth term is a linear expression in n i.e. its nth term is given by An + B, where A and S are constant and A is common difference.<\/p>\n<p>nth term of an AP : If a is the first term, d is common difference and l is the last term of an AP then<\/p>\n<ul>\n<li>nth term is given by a<sub>n<\/sub> = a + (n \u2013 1)d.<\/li>\n<li>nth term of an AP from the last term is a\u2019<sub>n<\/sub> =a<sub>n<\/sub> \u2013 (n \u2013 1)d.<\/li>\n<li>a<sub>n<\/sub> + a\u2019<sub>n<\/sub> = constant<\/li>\n<li>Common difference of an AP i.e. d = a<sub>n<\/sub> \u2013 a<sub>n-1<\/sub>,\u2200 n &gt; 1.<\/li>\n<\/ul>\n<p>If a constant is added or subtracted from each term of an AR then the resulting sequence is an AP with same common difference.<\/p>\n<p>If each term of an AP is multiplied or divided by a non-zero constant, then the resulting sequence is also an AP.<\/p>\n<p>If a, b and c are three consecutive terms of an A.P then 2b = a + c.<\/p>\n<p>Any three terms of an AP can be taken as (a \u2013 d), a, (a + d) and any four terms of an AP can be taken as (a \u2013 3d), (a \u2013 d), (a + d), (a + 3d)<\/p>\n<p><strong>Sum of n Terms of an AP<\/strong><br \/>\nSum of n terms of an AP is given by<br \/>\nS<sub>n<\/sub> = \\(\\frac { n }{ 2 }\\) [2a + (n \u2013 1)d] = \\(\\frac { n }{ 2 }\\) (a<sub>1<\/sub>+ a<sub>n<\/sub>)<\/p>\n<p>A sequence is an AP If the sum of n terms is of the form An<sup>2<\/sup> + Bn, where A and B are constant and A = half of common difference i.e. 2A = d.<\/p>\n<p>a<sub>n<\/sub> =S<sub>n<\/sub> \u2013 S<sub>n-1<\/sub><\/p>\n<p><strong>Arithmetic Mean<\/strong><br \/>\nIf a, A and b are in A.P then A = \\(\\frac { a+b }{ 2 }\\) is called the arithmetic mean of a and b.<\/p>\n<p>If a<sub>1<\/sub>, a<sub>2<\/sub>, a<sub>3<\/sub>,\u2026\u2026a<sub>n<\/sub> are n numbers, then their arithmetic mean is given by<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-71295\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png\" alt=\"Sequences and Series Class 11 Notes Maths Chapter 9\" width=\"459\" height=\"291\" \/><\/p>\n<p><strong>Geometric Progression (GP)<\/strong><br \/>\nA sequence in which the ratio of two consecutive terms is constant is called geometric progression. The constant ratio is called common ratio(r).<br \/>\ni.e. r = \\(\\frac { { a }_{ n }+1 }{ { a }_{ n } }\\), \u2200 n &gt; 1<\/p>\n<p><strong>Properties of Geometric Progression<br \/>\n<\/strong>If a is the first term and r is the common ratio, then the general term or nth term of GP is a<sub>n<\/sub> =ar<sup>n-1<\/sup><\/p>\n<p>nth term of a GP from the end is a\u2019<sub>n<\/sub> = \\(\\frac { 1 }{ { r }^{ n-1 } }\\), l = last term<\/p>\n<p>If all the terms of GP be multiplied or divided by same non-zero constant, then the resulting sequence is a GP with the same common ratio.<\/p>\n<p>The reciprocal terms of a given GP form a GP.<\/p>\n<p>If each term of a GP be raised to some power, the resulting sequence also forms a GP<\/p>\n<p>If a, b and c are three consecutive terms of a GP then b<sup>2<\/sup> = ac.<\/p>\n<p>Any three terms can be taken in GP as \\(\\frac { a }{ r }\\), a and ar and any four terms can be taken in GP as \\(\\frac { a }{ { r }^{ 3 } }\\), \\(\\frac { a }{ r }\\), ar and ar<sup>3<\/sup>.<\/p>\n<p><strong>Sum of n Terms of a G.P<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-71296\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-2-1.png\" alt=\"Sequences and Series Class 11 Notes Maths Chapter 9\" width=\"365\" height=\"263\" \/><\/p>\n<p><strong>Geometric Mean (GM)<\/strong><br \/>\nIf a, G and b are in GR then G is called the geometric mean of a and b and is given by G = \u221a(ab).<\/p>\n<p>If a,G<sub>1<\/sub>, G<sub>2<\/sub>, G<sub>3<\/sub>,\u2026.. G<sub>n<\/sub>, b are in GP then G<sub>1<\/sub>, G<sub>2<\/sub>, G<sub>3<\/sub>,\u2026\u2026G<sub>n<\/sub> are in GM\u2019s between a and b, then<br \/>\ncommon ratio r = \\(\\left( \\frac { b }{ a } \\right) ^{ \\frac { 1 }{ n+1 } }\\)<\/p>\n<p>If a<sub>1<\/sub>, a<sub>2<\/sub>, a<sub>3<\/sub>,\u2026, a<sub>n<\/sub> are n numbers are non-zero and non-negative, then their GM is given by<br \/>\nGM = (a<sub>1<\/sub> . a<sub>2<\/sub> . a<sub>3<\/sub> \u2026a<sub>n<\/sub>)<sup>1\/n<\/sup><\/p>\n<p>Product of n GM is G<sub>1<\/sub> \u00d7 G<sub>2<\/sub> \u00d7 G<sub>3<\/sub> \u00d7\u2026 \u00d7 G<sub>n<\/sub> =G<sub>n<\/sub> = \\(\\left( ab \\right) ^{ \\frac { n }{ 2 } }\\)<\/p>\n<p><strong>Important Results on the Sum of Special Sequences<\/strong><br \/>\nSum of first n natural numbers is<br \/>\n\u03a3n = 1 + 2 + 3 +\u2026 + n = \\(\\frac { n(n+1) }{ 2 }\\)<\/p>\n<p>Sum of squares of first n natural numbers is<br \/>\n\u03a3n<sup>2<\/sup> = 1<sup>2<\/sup> + 2<sup>2<\/sup> + 3<sup>2<\/sup> + \u2026 + n<sup>2<\/sup> = \\(\\frac { n(n+1)(2n+1) }{ 6 }\\)<\/p>\n<p>Sum of cubes of first n natural numbers is<br \/>\n\u03a3n<sup>3<\/sup> = 1<sup>3<\/sup> + 2<sup>3<\/sup> + 3<sup>3<\/sup> + .. + n<sup>3<\/sup> = \\(\\left( \\frac { n\\left( n+1 \\right) \\left( 2n+1 \\right) }{ 6 } \\right) ^{ 2 }\\)<\/p>\n<div class=\"featured-snippet\"><\/div>\n<h5><\/h5>\n","protected":false},"excerpt":{"rendered":"<p>Sequence A succession of numbers arranged in a definite order according to a given certain rule is called sequence. A [&hellip;]<\/p>\n","protected":false},"author":31,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"Sequences and Series Class 11 Notes Maths Chapter 9","_yoast_wpseo_metadesc":"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a'n = ...","custom_permalink":"study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/"},"categories":[92,93,13,21],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Sequences and Series Class 11 Notes Maths Chapter 9<\/title>\n<meta name=\"description\" content=\"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a&#039;n = ...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sequences and Series Class 11 Notes Maths Chapter 9\" \/>\n<meta property=\"og:description\" content=\"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a&#039;n = ...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-01-13T12:29:36+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Vipin2\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Sequences and Series Class 11 Notes Maths Chapter 9","description":"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a'n = ...","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/","og_locale":"en_US","og_type":"article","og_title":"Sequences and Series Class 11 Notes Maths Chapter 9","og_description":"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a'n = ...","og_url":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2022-01-13T12:29:36+00:00","og_image":[{"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"Vipin2","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#primaryimage","inLanguage":"en-US","url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png?v=1642485980","contentUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png?v=1642485980","width":459,"height":291},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/","name":"Sequences and Series Class 11 Notes Maths Chapter 9","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"primaryImageOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#primaryimage"},"datePublished":"2022-01-13T12:29:36+00:00","dateModified":"2022-01-13T12:29:36+00:00","description":"CBSE Class 11 Maths Notes Chapter 9 Sequences and Series \u00b7 nth term is given by an = a + (n \u2013 1)d. \u00b7 nth term of an AP from the last term is a'n = ...","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Sequences and Series Class 11 Notes Maths Chapter 9"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/adefc332a5319edb4ac9abf1e2e8a189"},"headline":"Sequences and Series Class 11 Notes Maths Chapter 9","datePublished":"2022-01-13T12:29:36+00:00","dateModified":"2022-01-13T12:29:36+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#webpage"},"wordCount":753,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/study-materials\/sequences-and-series\/class-11-notes-maths\/chapter-9\/#primaryimage"},"thumbnailUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/Sequences-and-Series-Class-11-Notes-Maths-Chapter-9-1-1.png","articleSection":["CBSE Notes","Important Questions","Maths","Study Materials"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/adefc332a5319edb4ac9abf1e2e8a189","name":"Vipin2","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/fcfa86ca5a6e50482cab04ae92584cc1?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/fcfa86ca5a6e50482cab04ae92584cc1?s=96&d=mm&r=g","caption":"Vipin2"},"url":"https:\/\/infinitylearn.com\/surge\/author\/vipin2\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/28134"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/31"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=28134"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/28134\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=28134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=28134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=28134"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=28134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}