If the sum of the coefficients of all even po | vedclass

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(A) Let $P(x) = (1+x+x^{2}+\ldots+x^{2n})(1-x+x^{2}-x^{3}+\ldots+x^{2n}) = \sum_{k=0}^{4n} a_k x^k$. The sum of coefficients of even powers is given b

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$30$

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(A) Let $P(x) = (1+x+x^{2}+\ldots+x^{2n})(1-x+x^{2}-x^{3}+\ldots+x^{2n}) = \sum_{k=0}^{4n} a_k x^k$. The sum of coefficients of even powers is given by $\frac{P(1) + P(-1)}{2}$. First,calculate $P(1)$: $P(1) = (1+1+1^{2}+\ldots+1^{2n})(1-1+1^{2}-1^{3}+\ldots+1^{2n}) = (2n+1)(1) = 2n+1$. Next,calculate $P(-1)$: $P(-1) = (1-1+1-1+\ldots+1)(1-(-1)+(-1)^{2}-(-1)^{3}+\ldots+(-1)^{2n}) = (1)(1+1+1+\ldots+1) = (1)(2n+1) = 2n+1$. Sum of coefficients of even powers $= \frac{(2n+1) + (2n+1)}{2} = 2n+1$. Given $2n+1 = 61$,we get $2n = 60$,so $n = 30$.

If the sum of the coefficients of all even powers of $x$ in the product $(1+x+x^{2}+\ldots+x^{2n})(1-x+x^{2}-x^{3}+\ldots+x^{2n})$ is $61$,then $n$ is equal to

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