If $a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . + a_{2 n}^{2} = k a_{n}$ , then $k$ equals

1

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2

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\frac{1}{2}

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0

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Solution

The correct option is A $1$
Replacing $x$ by $- 1 / x$ in $(1)$ , we get
${(1 - \frac{1}{x} + \frac{1}{x^{2}})}^{n} = \sum_{r = 0}^{2 n} {(- 1)}^{r} \frac{a_{r}}{x^{r}}$
Note that
$a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . + a_{2 n}^{2} =$ coefficient of the constant term in
$[a_{0} + a_{1} x + . . . . + a_{2 n} x^{2 n}] [a_{0} + \frac{a_{1}}{x} + \frac{a_{2}}{x^{2}} - \frac{a_{3}}{x^{3}} + . . . . + \frac{a_{2 n}}{x^{2 n}}]$
$=$ coefficient of the constant term in
${(x^{2} + x + 1)}^{n} {(1 - \frac{1}{x} + \frac{1}{x^{2}})}^{n}$
$=$ coefficient of the constant term in
$= \frac{{(x^{2} + x + 1)}^{n} {(x^{2} - x + 1)}^{n}}{x^{2 n}}$
$=$ coefficient of $x^{2 n}$ in ${[{{(x}^{2} + 1)}^{2} - x^{2}]}^{n}$
$=$ coefficient of $x^{2 n}$ in ${(1 + x^{2} + x^{4})}^{n}$
$=$ coefficient of $y^{n}$ in ${(1 + y + y^{2})}^{n}$
$= a_{n}$

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