求和（x+1/x)^2+(x^2+1/x^2)^2+.+(x^n+1/x^n)^2的值
问题描述：
求和（x+1/x)^2+(x^2+1/x^2)^2+.+(x^n+1/x^n)^2的值
答
求和（x+1/x)²+(x²+1/x²)²+.+(xⁿ+1/xⁿ)²的值
原式=(x²+x⁴+.+x²ⁿ)+(1/x²+1/x⁴+.+1/x²ⁿ)+2n
=x²(x²ⁿ-1)/(x²-1)+(1/x²)(1-1/x²ⁿ)/(1-1/x²)+2n
=x²(x²ⁿ-1)/(x²-1)+(1-1/x²ⁿ)/(x²-1)+2n
=[x^(2n+2)-x²+1-1/x²ⁿ]/(x²-1)+2n
=[x^(4n+2)-x^(2n+2)+x^2n-1]/[x²ⁿ(x²+1)]+2n