If the sum of the coefficients of all even powers of x in the product (1+x+x^2++x^2n)(1-x+…

Question

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

Answer: 30

Accepted Answer

Let $f(x)=(1+x+\cdots+x^{2n})(1-x+x^2-\cdots+x^{2n})=a_0+a_1x+\cdots$

At $x=1$: $f(1)=(2n+1)	imes1=2n+1=a_0+a_1+\cdots$ ...(i)

At $x=-1$: $f(-1)=1	imes(2n+1)=2n+1=a_0-a_1+a_2-\cdots$ ...(ii)

Adding: $4n+2=2(a_0+a_2+\cdots)=2	imes61=122\Rightarrow2n+1=61\Rightarrow n=30$

Solutions · 1