JEE Main · 2008 · 27 Apr 2008 · Offline · Binomial Theorem

Statement -1:

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}

.

Statement-2:

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}

(A)Statement -1 is false, Statement -2 is true
(B)Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement -1✓ Correct
(C)Statement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1.
(D)Statement - 1 is true, Statement - 2 is false.

Answer: (B)

Solution

Splitting the sum,

\sum_{r = 0}^{n} (r + 1) (r n) = \sum r (r n) + \sum (r n) = n 2^{n - 1} + 2^{n} = (n + 2) 2^{n - 1}

, so the first statement is true. The generating identity

\sum_{r = 0}^{n} (r + 1) (r n) x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}

is the second statement, which is also true, and putting

x = 1

in it reproduces the first statement exactly. The second therefore both holds and correctly explains the first.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option