If ${(1 + x + x^{2})}^{n} = (a_{0} + a_{1} . x + a_{2} x^{2} + a_{3} x^{3} + ...... + a_{2 n} . x^{2 n}),$ then
$a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + ..... + a_{2 n}^{2} =$

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$a_{n}$

$a_{n - 1}$

$- a_{n}$

answer is B.

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Detailed Solution

${(1 + x + x^{2})}^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + ....... + a_{2 n} x^{2 n} \to (i)$
${(1 - x + x^{2})}^{n} = a_{0} - a_{1} x + a_{2} x^{2} - a_{3} x^{3} + ....... + a_{2 n} x^{2 n}$ [getting by writing –x in the place of x in (i) ]
Replace ‘x’ with 1/x in the above equation, we get
${(1 - x + x^{2})}^{n} = a_{0} x^{2 n} - a_{1} x^{2 n - 1} + a_{2} x^{2 n - 2} - a_{3} x^{2 n - 3} + ..... + a_{2 n} x^{2 n}$ [getting by writing –x in the place of x in (i) ]
Replace ‘x’ with 1/x in the above equation, we get
${(1 + {x^{2}}^{n}) - x^{2}}^{n} = (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + ....... + a_{2 n} x^{2 n}) \times (a_{0} x^{2 n} - a_{1} x^{2 n - 1} + a_{2} x^{2 n - 2} - a_{3} x^{2 n - 3} + ... + a_{2 n})$

Coefficient of $x^{2 n}$ in R.H.S = ${a_{0}}^{2} - {a_{1}}^{2} + {a_{2}}^{2} - {a_{3}}^{2} + ...... + {a_{2 n}}^{2}$
L.H.S = ${(1 + x^{2} + x^{4})}^{n} = a_{0} + a_{1} x^{2} + a_{2} x^{4} + ...... + a_{n} x^{2 n} + ...... + a_{2 n} x^{4 n}$
$∴$ Coefficient of $x^{2 n} = a_{n}$
$∴ {a_{0}}^{2} - {a_{1}}^{2} + {a_{2}}^{2} - {a_{3}}^{2} + .... + {a_{2 n}}^{2} = a_{n}$