2 C_0 + 2^2 2 C_1 + 2^3 3 C_2 + .... + 2^ 11 11 C_ 10 = . . . - Vidyadip

MCQ

$2{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + .... + \frac{{{2^{11}}}}{{11}}{C_{10}}$ = . . .

✓
$\frac{{{3^{11}} - 1}}{{11}}$
B
$\frac{{{2^{11}} - 1}}{{11}}$
C
$\frac{{{{11}^3} - 1}}{{11}}$
D
$\frac{{{{11}^2} - 1}}{{11}}$

✓

Answer

Correct option: A.

$\frac{{{3^{11}} - 1}}{{11}}$

a
(a) We have ${(1 + x)^{10}} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_{10}}{x^{10}}$

Integrating both sides from $0$ to $2$, we get

$\frac{{{3^{11}} - 1}}{{11}} = 2{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + .... + \frac{{{2^{11}}}}{{11}}{C_{10}}$.

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