Solveeit Logo
Login

Question

Question: The value of 2(<sup>n</sup>C<sub>0</sub>) + \(\frac{3}{2}\) (<sup>n</sup>C<sub>1</sub>) + \(\frac{4}...

The value of 2(nC0) + 32\frac{3}{2} (nC1) + 43\frac{4}{3} (nC2) + 54\frac{5}{4} (nC3) ….. is –

A

2n(1−n)−1n+1\frac{2^{n}(1 - n) - 1}{n + 1}

B

2n(n+3)−1n+1\frac{2^{n}(n + 3) - 1}{n + 1}

C

2n−1n+1\frac{2^{n} - 1}{n + 1}

D

2n+2n−1\frac{2^{n} + 2}{n - 1}

Answer

2n(n+3)−1n+1\frac{2^{n}(n + 3) - 1}{n + 1}

Explanation

Solution

(1 + x)n = nC0 + nC1x + nC2x2 + nC3x3 … nCnxn

on Integrating

(1+x)n+1−1n+1\frac{(1 + x)^{n + 1} - 1}{n + 1} = nC0x +nC1x22\frac{nC_{1}x^{2}}{2}+ nC2x33\frac{nC_{2}x^{3}}{3}+ nC3x44\frac{nC_{3}x^{4}}{4} ……..

Multiplying with x & differentiating

ddx{x((1+x)n+1−1n+1)}\frac{d}{dx}\left\{ x\left( \frac{(1 + x)^{n + 1} - 1}{n + 1} \right) \right\} =

ddx{nC0x2+nC1x32+nC2x43+nC3x55..........}\frac{d}{dx}\left\{ nC_{0}x^{2} + \frac{nC_{1}x^{3}}{2} + \frac{nC_{2}x^{4}}{3} + \frac{nC_{3}x^{5}}{5}.......... \right\}

(1+x)n+1+x(n+1)(1+x)n−1n+1\frac{(1 + x)^{n + 1} + x(n + 1)(1 + x)^{n} - 1}{n + 1}= 2nC0x + 3nC1x22\frac{3^{n}C_{1}x^{2}}{2}+ 4nC2x33\frac{4^{n}C_{2}x^{3}}{3} ………

put x = 1

2n+1+(n+1)2n−1n+1\frac{2^{n + 1} + (n + 1)2^{n} - 1}{n + 1}= 2nC0 + 32\frac{3}{2} nC1 + 43nC2\frac{4}{3}^{n}C_{2}+ ….

= 2n(n+3)−1n+1\frac{2^{n}(n + 3) - 1}{n + 1}