{"id":7055,"date":"2021-10-22T15:58:37","date_gmt":"2021-10-22T10:28:37","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7055"},"modified":"2021-10-25T02:19:14","modified_gmt":"2021-10-24T20:49:14","slug":"let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/","title":{"rendered":"Let \\(a_n\\) be the nth term of an AP. If \\(\\sum_{r=1}^{100}\\) \\(a_{2r}\\) = \\(\\alpha\\) and \\(\\sum_{r=1}^{100}\\) \\(a_{2r-1}\\) = \\(\\beta\\), then the common difference of the AP is"},"content":{"rendered":"

Solution :<\/h2>\n

Given, \\(a_2 + a_4 + a_6 + …… + a_{200}\\) = \\(\\alpha\\)\u00a0 \u00a0 \u00a0 ………(i)<\/p>\n

and \\(a_1 + a_3 + a_5 + ….. + a_{199}\\) = \\(\\beta\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0………(ii)<\/p>\n

On subtracting equation (ii) from equation (i), we get<\/p>\n

(\\(a_2 – a_1\\)) + (\\(a_4 – a_3\\)) + ……… + (\\(a_{200} – a_{199}\\)) = \\(\\alpha\\) – \\(\\beta\\)<\/p>\n

\\(\\implies\\) d + d + ……..+ 100 times =\u00a0 \\(\\alpha\\) – \\(\\beta\\)<\/p>\n

\\(\\implies\\)\u00a0 100d = \\(\\alpha\\) – \\(\\beta\\)<\/p>\n

\\(\\therefore\\)\u00a0 \u00a0d = \\(\\alpha – \\beta\\over 100\\)<\/p>\n

\n

Similar Questions<\/h3>\n

Three positive integers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then the common ratio of GP is<\/a><\/p>\n

A man saves Rs 200 in each of the first three months of his service. In each of the subsequent months, his saving increases by Rs 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs 11040 after<\/a><\/p>\n

If 100 times the 100th term of an AP with non-zero common difference equal to the 50 times its 50th term, then the 150th term of AP is<\/a><\/p>\n

If x, y and z are in AP and \\(tan^{-1}x\\), \\(tan^{-1}y\\) and \\(tan^{-1}z\\) are also in AP, then<\/a><\/p>\n

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, \u2026\u2026. , is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : Given, \\(a_2 + a_4 + a_6 + …… + a_{200}\\) = \\(\\alpha\\)\u00a0 \u00a0 \u00a0 ………(i) and \\(a_1 + a_3 + a_5 + ….. + a_{199}\\) = \\(\\beta\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0………(ii) On subtracting equation (ii) from equation (i), we get (\\(a_2 – a_1\\)) + (\\(a_4 – a_3\\)) + ……… + (\\(a_{200} – …<\/p>\n

Let \\(a_n\\) be the nth term of an AP. If \\(\\sum_{r=1}^{100}\\) \\(a_{2r}\\) = \\(\\alpha\\) and \\(\\sum_{r=1}^{100}\\) \\(a_{2r-1}\\) = \\(\\beta\\), then the common difference of the AP is<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,57],"tags":[],"yoast_head":"\nLet \\(a_n\\) be the nth term of an AP. If \\(\\sum_{r=1}^{100}\\) \\(a_{2r}\\) = \\(\\alpha\\) and \\(\\sum_{r=1}^{100}\\) \\(a_{2r-1}\\) = \\(\\beta\\), then the common difference of the AP is<\/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:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let \\(a_n\\) be the nth term of an AP. 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If (sum_{r=1}^{100}) (a_{2r}) = (alpha) and (sum_{r=1}^{100}) (a_{2r-1}) = (beta), then the common difference of the AP is Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-22T10:28:37+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-24T20:49:14+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Let \\\\(a_n\\\\) be the nth term of an AP. 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