If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots .+x^

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

Professor PC Thomas Classes · Answer

(30) : Let $\left(1+x+x^{2}+\ldots+x^{2}{ }_{n}ight)(1-x$ $\left.+x^{2}-\ldots+x^{2}{ }_{n}ight)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{4 n} x^{4}{ }_{n}$ Putting $x=1$ in (i), we get $$\begin{equation*}(2 n+1) 1=a_{0}+a_{1}+a_{2}+\ldots+a_{4 n} 	ag{i} \end{equation*}$$ Putting $x=-1$ in (i), we get $$\begin{equation*}1(2 n+1)=a_{0}-a_{1}+a_{2}-\ldots+a_{4 n} 	ag{ii} \end{equation*}$$ Adding (ii) and (iii), we get $$\begin{equation*}4 n+2=2\left(a_{0}+a_{2}+\ldots+a_{4 n}ight)=2 	imes 61 	ag{iii} \end{equation*}$$ $\Rightarrow \quad 2 n+1=61 \Rightarrow n=30$\