If $| x | < 1$ then the coefficient of $x^{n}$ in 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 A $n$
$(1 + x + x^{2} + . . . . . x^{n})^{2} =^{2} C_{0} +^{2} C_{1} (x + x^{2} + . . . . x^{n}) +^{2} C_{2} (x + . . . . . x^{n})^{2}$
Coefficient of $x^{n} = 1 +$ coefficient of $x^{n}$ in $(x + . . . . . . . . x^{n})^{2}$
$= 1 + 1 +$ coefficient $x^{n}$ in $(x^{2} + . . . . . . x^{n})^{2}$
$1 + 1 + 1 +$ coefficient of $x^{n}$ in $(x^{3} + . . . . . . . x^{n})^{2}$
$= n +$ coefficient o f $x^{n}$ in $(x)^{2}$
$= n$
$A$ is correct.

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