The value of the sum of the series sum from r=0 to n of (r+1) * nCr is:

A. (n+2) 2^n-1 — The sum can be split into sum of r*nCr + sum of nCr. Using the identity r*nCr = n*n-1Cr-1, the sum becomes n*2^(n-1) + 2^n, which simplifies to (n+2)*2^(n-1).

JEE Main · MathematicsHard

$The value of the sum \sum_{r = 0}^{n} (r + 1) (r n) is:$

A. $(n + 2) 2^{n - 1}$
B. $(n + 1) 2^{n}$
C. $n 2^{n - 1}$
D. $(n + 2) 2^{n}$

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Correct answer: A — $(n + 2) 2^{n - 1}$

Solution

The sum is \sum r (r n) + \sum (r n) = n 2^{n - 1} + 2^{n} = 2^{n - 1} (n + 2) .

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The value of the sum ∑r=0n​(r+1)(rn​) is:

Solution

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$The value of the sum \sum_{r = 0}^{n} (r + 1) (r n) is:$