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Question: In the expansion of the following expression. \(1 + ( 1 + x ) +\) \(\frac{2^{n}}{n + 1}\)the coeffi...

In the expansion of the following expression.

1+(1+x)+1 + ( 1 + x ) + 2nn+1\frac{2^{n}}{n + 1}the coefficient of 2n−1n+1\frac{2^{n} - 1}{n + 1} is.

A

(n+1)(n + 1)

B

C02−C13+C24−C35+\frac{C_{0}}{2} - \frac{C_{1}}{3} + \frac{C_{2}}{4} - \frac{C_{3}}{5} +

C

1n+1\frac{1}{n + 1}

D

None of these

Answer

(n+1)(n + 1)

Explanation

Solution

The expression being in G. P.

1+y=(1−x)−3⇒1−x=(1+y)−1/31 + y = (1 - x)^{- 3} \Rightarrow 1 - x = (1 + y)^{- 1/3}

x=1−(1+y)−1/3x = 1 - (1 + y)^{- 1/3}

[1+x2−x]−1=11+x2−x×(1+x2+x)(1+x2+x)\lbrack\sqrt{1 + x^{2}} - x\rbrack^{- 1} = \frac{1}{\sqrt{1 + x^{2}} - x} \times \frac{(\sqrt{1 + x^{2}} + x)}{(\sqrt{1 + x^{2}} + x)}The coefficient of x k in E

= The coefficient of =1+x2+x1+x2−x2=x+1+x2=x+(1+x2)1/2= \frac{\sqrt{1 + x^{2}} + x}{1 + x^{2} - x^{2}} = x + \sqrt{1 + x^{2}} = x + (1 + x^{2})^{1/2}in =x+1+12x2+12(−12)x42!+...= x + 1 + \frac{1}{2}x^{2} + \frac{1}{2}\left( - \frac{1}{2} \right)\frac{x^{4}}{2!} + ... (1+x)n(1 + x)^{n}.