\nLet the sum be $S$.
\n$S = 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\nMultiply S by the common ratio, $r=2$:
\n$2S = 1\\cdot2^2 + 2\\cdot2^3 + 3\\cdot2^4 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\nSubtract the first equation from the second:
\n$2S – S = (1\\cdot2^2 + 2\\cdot2^3 + \\dots + 10\\cdot2^{11}) – (1\\cdot2^1 + 2\\cdot2^2 + \\dots + 10\\cdot2^{10})$
\n$S = -1\\cdot2^1 + (2-1)2^2 + (3-2)2^3 + \\dots + (10-9)2^{10} + 10\\cdot2^{11}$
\n$S = -2 + 1\\cdot2^2 + 1\\cdot2^3 + \\dots + 1\\cdot2^{10} + 10\\cdot2^{11}$
\n$S = (2^2 + 2^3 + \\dots + 2^{10}) – 2 + 10\\cdot2^{11}$
\nThe terms in the parenthesis form a geometric series with first term $a = 2^2 = 4$, common ratio $r=2$, and number of terms $n = 10-2+1 = 9$.
\nThe sum of this geometric series is $G = a \\frac{r^n – 1}{r-1} = 4 \\frac{2^9 – 1}{2-1} = 4 (2^9 – 1) = 2^2 (2^9 – 1) = 2^{11} – 4$.
\nSubstitute this back into the expression for S:
\n$S = (2^{11} – 4) – 2 + 10\\cdot2^{11}$
\n$S = 2^{11} – 6 + 10\\cdot2^{11}$
\n$S = (1 + 10)2^{11} – 6$
\n$S = 11\\cdot2^{11} – 6$.<\/p>\nHold on, let’s re-calculate the subtraction step correctly.
\n$S = 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\n$2S = \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\n$2S – S = (1\\cdot2^2 – 2\\cdot2^2) + (2\\cdot2^3 – 3\\cdot2^3) + \\dots + (9\\cdot2^{10} – 10\\cdot2^{10}) + 10\\cdot2^{11} – 1\\cdot2^1$
\nThis is incorrect. The terms should align:
\n$S = \\quad 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\n$2S = \\quad \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\nSubtracting S from 2S:
\n$S = (10\\cdot2^{11}) – (1\\cdot2^1) – [(2-1)2^2 + (3-2)2^3 + \\dots + (10-9)2^{10}]$
\n$S = 10\\cdot2^{11} – 2 – [2^2 + 2^3 + \\dots + 2^{10}]$
\nThe geometric series is $2^2 + 2^3 + \\dots + 2^{10}$. First term $a=2^2=4$, ratio $r=2$, number of terms $n=9$.
\nSum $G = 4 \\frac{2^9 – 1}{2-1} = 4(512 – 1) = 4 \\times 511 = 2044$.
\nLet’s re-calculate $G$ using the formula $a(r^n-1)\/(r-1)$ where $a=2$, $n=10$ for $2^1 + … + 2^{10}$, then subtract the first term $2^1$.
\n$G’ = 2^1 + 2^2 + \\dots + 2^{10} = 2 \\frac{2^{10} – 1}{2-1} = 2(1024 – 1) = 2 \\times 1023 = 2046$.
\nThe sum $2^2 + 2^3 + \\dots + 2^{10} = G’ – 2^1 = 2046 – 2 = 2044$.
\n$S = 10\\cdot2^{11} – 2 – 2044 = 10\\cdot2^{11} – 2046$.
\n$10 \\cdot 2^{11} = 10 \\cdot 2048 = 20480$.
\n$S = 20480 – 2046 = 18434$.<\/p>\n
Now check the options:
\nA) $9\\times2^{11} + 2 = 9 \\times 2048 + 2 = 18432 + 2 = 18434$. Matches.
\nB) $10\\times2^{11} + 2 = 10 \\times 2048 + 2 = 20480 + 2 = 20482$.
\nC) $10\\times2^{11} – 2 = 10 \\times 2048 – 2 = 20480 – 2 = 20478$.
\nD) $9\\times2^{11} – 2 = 9 \\times 2048 – 2 = 18432 – 2 = 18430$.<\/p>\n
My original calculation of the geometric series part in the subtraction was correct:
\n$S = 10 \\cdot 2^{11} – (2^1 + 2^2 + \\dots + 2^{10})$
\nThe geometric series $2^1 + 2^2 + \\dots + 2^{10}$ has $a=2, r=2, n=10$.
\nSum is $2 \\frac{2^{10}-1}{2-1} = 2(1024-1) = 2(1023) = 2046$.
\n$S = 10 \\cdot 2^{11} – 2046$.
\n$S = 10 \\cdot 2048 – 2046 = 20480 – 2046 = 18434$.
\nThis matches $9 \\cdot 2^{11} + 2$.<\/p>\n
Let’s re-do the AGP subtraction step structure:
\n$S = 1\\cdot2 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\n$2S = \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\nSubtracting the first from the second, aligning terms vertically:
\n$S = (2\\cdot2^2 – 1\\cdot2^2) + (3\\cdot2^3 – 2\\cdot2^3) + \\dots + (10\\cdot2^{10} – 9\\cdot2^{10}) + 10\\cdot2^{11} – 1\\cdot2^1$ — This is wrong alignment
\nCorrect alignment for subtraction:
\n$2S = \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\n$S = 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$<\/p>\n
$2S – S = (10\\cdot2^{11}) + (9\\cdot2^{10} – 10\\cdot2^{10}) + \\dots + (2\\cdot2^2 – 3\\cdot2^2) + (1\\cdot2^1 – 2\\cdot2^1)$ — Also wrong alignment.<\/p>\n
Correct method:
\n$S = 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\n$2S = \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$<\/p>\n
$2S – S = (10\\cdot2^{11}) – (1\\cdot2^1) + (1\\cdot2^2 – 2\\cdot2^2) + (2\\cdot2^3 – 3\\cdot2^3) + \\dots + (9\\cdot2^{10} – 10\\cdot2^{10})$ This is also not right.<\/p>\n
The general formula for $\\sum_{k=1}^n k r^k$ is $\\frac{nr^{n+2} – (n+1)r^{n+1} + r}{(r-1)^2}$.
\nHere $n=10$, $r=2$.
\n$S = \\frac{10 \\cdot 2^{12} – (10+1)2^{11} + 2}{(2-1)^2} = \\frac{10 \\cdot 2^{12} – 11 \\cdot 2^{11} + 2}{1^2}$
\n$S = 10 \\cdot 2 \\cdot 2^{11} – 11 \\cdot 2^{11} + 2$
\n$S = 20 \\cdot 2^{11} – 11 \\cdot 2^{11} + 2$
\n$S = (20 – 11) \\cdot 2^{11} + 2$
\n$S = 9 \\cdot 2^{11} + 2$.
\nThis matches option A and confirms the previous calculation result.<\/p>\n
The error in manual subtraction breakdown was in the terms. It should be:
\n$S = 1\\cdot2 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + 10\\cdot2^{10}$
\n$2S = \\quad \\quad 1\\cdot2^2 + 2\\cdot2^3 + \\dots + 9\\cdot2^{10} + 10\\cdot2^{11}$
\n$2S – S = (10\\cdot2^{11}) + (2\\cdot2^2 – 1\\cdot2^2) + (3\\cdot2^3 – 2\\cdot2^3) + \\dots + (10\\cdot2^{10} – 9\\cdot2^{10}) – 1\\cdot2^1$
\nThis is still not quite right. The terms are offset.
\n$S = 1\\cdot2^1 + 2\\cdot2^2 + 3\\cdot2^3 + \\dots + (n-1)r^{n-1} + n r^n$
\n$rS = \\quad \\quad 1\\cdot r^2 + 2\\cdot r^3 + \\dots + (n-1)r^n + n r^{n+1}$
\n$rS – S = n r^{n+1} – 1\\cdot r^1 – [(2-1)r^2 + (3-2)r^3 + \\dots + (n-(n-1))r^n]$
\n$(r-1)S = n r^{n+1} – r – [r^2 + r^3 + \\dots + r^n]$
\nThe geometric series in the bracket is $r^2 + \\dots + r^n$. First term $r^2$, ratio $r$, number of terms $n-1$.
\nSum is $r^2 \\frac{r^{n-1}-1}{r-1}$.
\n$(r-1)S = n r^{n+1} – r – r^2 \\frac{r^{n-1}-1}{r-1}$. For $r=2$:
\n$S = n 2^{n+1} – 2 – 4 \\frac{2^{n-1}-1}{1} = n 2^{n+1} – 2 – 4(2^{n-1}-1) = n 2^{n+1} – 2 – 2^2 2^{n-1} + 4 = n 2^{n+1} – 2^{n+1} + 2$.
\n$S = (n-1)2^{n+1} + 2$.
\nWith $n=10$:
\n$S = (10-1)2^{10+1} + 2 = 9\\cdot2^{11} + 2$.
\nThis confirms the formula derived from the difference method.<\/p>\n<\/section>\n","protected":false},"excerpt":{"rendered":"
What is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + …… + 10\u00d72\u00b9\u2070 [amp_mcq option1=”9\u00d72\u00b9\u00b9 + 2″ option2=”10\u00d72\u00b9\u00b9 + 2″ option3=”10\u00d72\u00b9\u00b9 – 2″ option4=”9\u00d72\u00b9\u00b9 – 2″ correct=”option1″] This question was previously asked in UPSC CISF-AC-EXE – 2021 Download PDFAttempt Online The value of the sum $1\\times2 + 2\\times2^2 … <\/p>\n
Detailed SolutionWhat is the value of the following sum ?
\n1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1089],"tags":[1110,1102],"class_list":["post-92926","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1110","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nWhat is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..<\/title>\n<meta name=\"description\" content=\"The value of the sum $1times2 + 2times2^2 + 3times2^3 + ...... + 10times2^{10}$ is $9times2^{11} + 2$. This is an Arithmetico-Geometric Series (AGP) of the form $sum_{k=1}^n k r^k$. The sum of such a series can be found using a standard technique involving multiplying the series by the common ratio and subtracting the original series.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2\u00b2-3x2\u00b3-4x2\u2074\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..\" \/>\n<meta property=\"og:description\" content=\"The value of the sum $1times2 + 2times2^2 + 3times2^3 + ...... + 10times2^{10}$ is $9times2^{11} + 2$. This is an Arithmetico-Geometric Series (AGP) of the form $sum_{k=1}^n k r^k$. The sum of such a series can be found using a standard technique involving multiplying the series by the common ratio and subtracting the original series.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2\u00b2-3x2\u00b3-4x2\u2074\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:36:55+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"What is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..","description":"The value of the sum $1times2 + 2times2^2 + 3times2^3 + ...... + 10times2^{10}$ is $9times2^{11} + 2$. This is an Arithmetico-Geometric Series (AGP) of the form $sum_{k=1}^n k r^k$. The sum of such a series can be found using a standard technique involving multiplying the series by the common ratio and subtracting the original series.","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:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2\u00b2-3x2\u00b3-4x2\u2074\/","og_locale":"en_US","og_type":"article","og_title":"What is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..","og_description":"The value of the sum $1times2 + 2times2^2 + 3times2^3 + ...... + 10times2^{10}$ is $9times2^{11} + 2$. This is an Arithmetico-Geometric Series (AGP) of the form $sum_{k=1}^n k r^k$. The sum of such a series can be found using a standard technique involving multiplying the series by the common ratio and subtracting the original series.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2\u00b2-3x2\u00b3-4x2\u2074\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:36:55+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2%c2%b2-3x2%c2%b3-4x2%e2%81%b4\/","url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-value-of-the-following-sum-1x2-2x2%c2%b2-3x2%c2%b3-4x2%e2%81%b4\/","name":"What is the value of the following sum ? 1\u00d72 + 2\u00d72\u00b2 + 3\u00d72\u00b3 + 4\u00d72\u2074 + ..","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:36:55+00:00","dateModified":"2025-06-01T11:36:55+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The value of the sum $1\\times2 + 2\\times2^2 + 3\\times2^3 + ...... + 10\\times2^{10}$ is $9\\times2^{11} + 2$. 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