If $n$ is a positive integer, then the coefficient of $x^{n}$ in the expansion of $\frac{(1 + x)^{n}}{1 - x}$ is

(n + 1) 2^{n}

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2^{n}

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2^{n - 1}

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n {.2}^{n}

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Solution

The correct option is B $2^{n}$
To find the coefficient of $x^{n}$ in the expansion of $\frac{(1 + x)^{n}}{1 - x}$ .

This can be written as $(1 + x)^{n} (1 - x)^{- 1}$

$= ((\frac{n}{0}) + (\frac{n}{1}) x + (\frac{n}{2}) x^{2} + (\frac{n}{3}) x^{3} + . . . . . . . . . . . . + (\frac{n}{n}) x^{n}) (1 - x + x^{2} - x^{3} + x^{4} - . . . . . . . . .)$

On Multiplying this, we can find the coefficient of $x^{n}$ as

Summation of $= C_{0} + C_{1} + C_{2} + . . . . . . . . . . . . C_{n}$ $= 2^{n}$

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