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Binomial Theorem for Any Index

If | x |<1, t...

Question

If $| x | < 1$ ,then the coefficient of $x^{n}$ in the expansion of $(1 + x + x^{2} + x^{3} + . . . .)^{2}$ is

n

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n - 1

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n + 2

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n + 1

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Solution

The correct option is D $n + 1$
As $1 + x + x^{2} + x^{3} + \dots = (1 - x)^{- 1}$
$∴ (1 + x + x^{2} + . . . . .)^{2} = (1 - x)^{- 2}$
$= 1 + 2 x + 3 x^{2} + . . . . . .$
Therefore, coefficient of $x^{n}$ is $n + 1$ .

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