In the expansion of ( 1 + x 1 − x ) 2 , the coefficient of x n Options: A . &nbsp 4n B . &nbsp 4n-3 C . &nbsp 4n+1 D . &nbsp None of these

In the expansion of (1+x1−x)2, the coefficient of xn

Question -

In the expansion of $(\frac{1 + x}{1 - x})^{2}$ , the coefficient of $x^{n}$

Options:

A . 4n

B . 4n-3

C . 4n+1

D . None of these

Answer: Option A
:
A

Given term can be written as $(1 + x)^{2}$ $(1 - x)^{- 2}$

= $(1 + 2 x + x^{2}) [1 + 2 x + 3 x^{2} + . . . . . . . + (n - 1) x^{n - 2} + n x^{n - 1} + (n + 1) x^{n} + . . . . . . . . . . .]$

= $x^{n} (n + 1 + 2 n + n - 1)$ +..........

Therefore coefficient of $x^{n}$ is 4n.

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