[JEE Advanced 1996] if 1 x x 2 n a 0 a 1x a 2n x 2n
If $(1+x+x^2)^n=a_0+a_1x+\cdots+a_{2n}x^{2n}$, show that $a_0^2-a_1^2+a_2^2-\cdots+a_{2n}^2=a_n$.
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· 2d ago
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Answer: Proved $=a_n$.
$\sum(-1)^ka_k^2$ is the coefficient of $x^{2n}$ in $(1+x+x^2)^n(1-x+x^2)^n=(1+x^2+x^4)^n$, which equals $a_n$ (the central coefficient of $(1+y+y^2)^n$ with $y=x^2$).
JEE Advanced 1996 · Binomial Theorem — verified solution by the Vidaara Team.
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