If the sum of the coefficients of all

Question:

If the sum of the coefficients of all even powers of $x$ in the product

$\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ is 61 , then $n$ is equal to___________.

Solution:

Let $\left(1-x+x^{2} \ldots . . x^{2 n}\right)\left(1+x+x^{2} \ldots . . x^{2 n}\right)$

$=a_{0}+a_{1} x+a_{2} x^{2}+\ldots . .$

put $x=1$

$1(2 n+1)=a_{0}+a_{1}+a_{2}+\ldots . . a_{2 n}$.......(1)

put $x=-1$

$(2 n+1) \times 1=a_{0}-a_{1}+a_{2}+\ldots \ldots a_{2 n} \quad \ldots$ (2)

Adding (1) and (2), we get,

$4 n+2=2\left(a_{0}+a_{2}+\ldots \ldots\right)=2 \times 61$

$\Rightarrow 2 n+1=61 \Rightarrow n=30$

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