Find the middle term in the expansion of $(1 - 2 x + x^{2})^{n}$

\frac{(2 n)!}{(n!)^{2}} (- 1)^{n} x^{n}

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\frac{(2 n)!}{(n!)} (- 1)^{n} x^{2} n

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\frac{(2 n)!}{(n!)^{2}} x^{n}

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None of these

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Solution

The correct option is A $\frac{(2 n)!}{(n!)^{2}} (- 1)^{n} x^{n}$
Given expression $(1 - 2 x + x^{2})^{n}$

$\Rightarrow (1 - 2 x + x^{2})^{n} = [(1 - x)^{2}]^{n} = (1 - x)^{2 n}$

Middle term ${(\frac{2 n}{2} + 1)}^{t h} \Rightarrow (n + 1)^{t h}$ term

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

$= \frac{2 n!}{(n!)^{2}} (1)^{n} (- x)^{n}$

Middle term is $\frac{(2 n!)}{(n!)^{2}} (- 1)^{n} x^{n}$

Suggest Corrections

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