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Greatest Binomial Coefficients

The coefficie...

Question

The coefficient of $x^{n}$ , where $n$ is any positive integer, in the expansion of ${(1 + 2 x + 3 x^{2} + \dots \infty)}^{\frac{1}{2}}$ is

1

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\frac{n + 1}{2}

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2 n + 1

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n + 1

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Solution

The correct option is B $1$
We know that
${(1 - x)}^{- 2} = 1 + 2 x + 3 x^{2} + \dots \infty$
$\Rightarrow {[{(1 - x)}^{- 2}]}^{1 / 2} = {(1 + 2 x + 3 x^{2} + \dots \infty)}^{1 / 2}$
$\Rightarrow {(1 - x)}^{- 1} = {(1 + 2 x + 3 x^{2} + \dots \infty)}^{1 / 2}$
$\Rightarrow 1 + x + x^{2} + \dots \infty = {(1 + 2 x + 3 x^{2} + \dots \infty)}^{1 / 2}$
Hence, coefficient of $x^{n}$ is $1$ .

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