{"id":7037,"date":"2021-10-22T15:46:32","date_gmt":"2021-10-22T10:16:32","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7037"},"modified":"2021-10-25T02:23:42","modified_gmt":"2021-10-24T20:53:42","slug":"if-109-2111108-3112107-10119-k109-then-k-is-equal-to","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/","title":{"rendered":"If \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + …… + \\(10(11)^9\\) = \\(K(10)^9\\), then k is equal to"},"content":{"rendered":"

Solution :<\/h2>\n

\\(K(10)^9\\) = \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + …… + \\(10(11)^9\\)<\/p>\n

K = 1 + 2\\(({11\\over 10})\\)\u00a0 + 3\\(({11\\over 10})^2\\) + ….. + 10\\(({11\\over 10})^9\\)\u00a0 \u00a0 \u00a0 \u00a0……(i)<\/p>\n

\\(({11\\over 10})\\)K = 1\\(({11\\over 10})\\) + 2\\(({11\\over 10})^2\\) + 3\\(({11\\over 10})^3\\) + ….. + 10\\(({11\\over 10})^{10}\\)\u00a0 \u00a0 \u00a0 \u00a0…..(ii)<\/p>\n

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

K\\((1 – {11\\over 10})\\) = 1 + \\(({11\\over 10})\\) + \\(({11\\over 10})^2\\) + …. + \\(({11\\over 10})^9\\) – 10\\(({11\\over 10})^{10}\\)<\/p>\n

\\(\\implies\\) K\\(({10 – 11\\over 10})\\) = \\(1[({11\\over 10})^{10} – 1]\\over ({11\\over 10} – 1)\\) – 10\\(({11\\over 10})^{10}\\)<\/p>\n

\\(\\implies\\) – K = 10[10\\(({11\\over 10})^{10}\\) – 10 – 10\\(({11\\over 10})^{10}\\) ]<\/p>\n

\\(\\implies\\) K = 100<\/p>\n

\n

Similar Questions<\/h3>\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<\/a><\/p>\n

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, \u2026\u2026. , is<\/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

Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + \u2026\u2026<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : \\(K(10)^9\\) = \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + …… + \\(10(11)^9\\) K = 1 + 2\\(({11\\over 10})\\)\u00a0 + 3\\(({11\\over 10})^2\\) + ….. + 10\\(({11\\over 10})^9\\)\u00a0 \u00a0 \u00a0 \u00a0……(i) \\(({11\\over 10})\\)K = 1\\(({11\\over 10})\\) + 2\\(({11\\over 10})^2\\) + 3\\(({11\\over 10})^3\\) + ….. + 10\\(({11\\over 10})^{10}\\)\u00a0 \u00a0 \u00a0 \u00a0…..(ii) On subtracting equation (ii) from (i), …<\/p>\n

If \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + …… + \\(10(11)^9\\) = \\(K(10)^9\\), then k is equal to<\/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":"\nIf \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + ...... + \\(10(11)^9\\) = \\(K(10)^9\\), then k is equal to<\/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\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + ...... + \\(10(11)^9\\) = \\(K(10)^9\\), then k is equal to\" \/>\n<meta property=\"og:description\" content=\"Solution : (K(10)^9) = ((10)^9) + (2(11)^1(10)^8) + (3(11)^2(10)^7) + …… + (10(11)^9) K = 1 + 2(({11over 10}))\u00a0 + 3(({11over 10})^2) + ….. + 10(({11over 10})^9)\u00a0 \u00a0 \u00a0 \u00a0……(i) (({11over 10}))K = 1(({11over 10})) + 2(({11over 10})^2) + 3(({11over 10})^3) + ….. + 10(({11over 10})^{10})\u00a0 \u00a0 \u00a0 \u00a0…..(ii) On subtracting equation (ii) from (i), … If ((10)^9) + (2(11)^1(10)^8) + (3(11)^2(10)^7) + …… + (10(11)^9) = (K(10)^9), then k is equal to Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-22T10:16:32+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-24T20:53:42+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\" 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from (i), … If ((10)^9) + (2(11)^1(10)^8) + (3(11)^2(10)^7) + …… + (10(11)^9) = (K(10)^9), then k is equal to Read More »","og_url":"https:\/\/mathemerize.com\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/","og_site_name":"Mathemerize","article_published_time":"2021-10-22T10:16:32+00:00","article_modified_time":"2021-10-24T20:53:42+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/if-109-2111108-3112107-10119-k109-then-k-is-equal-to\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"If \\((10)^9\\) + \\(2(11)^1(10)^8\\) + \\(3(11)^2(10)^7\\) + …… + \\(10(11)^9\\) = \\(K(10)^9\\), then k is 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