Find the sum to `n` terms of the sequence `(x+1//x)^2,(x^2+1//x)^2,(x^3+1//x)^2, ,`

Explanation

Let `S_(n)` denote the sum to n terms of the given sequence Then,
`S_(n)=(x+1/x)^(2)+(x^(2)+1/(x^(2)))^(2)+(x^(3)+1/x^3)^(2)+…+(x^(n)+1/(x^(n)))^(2)`
`=(x^(2)+1/(x^(2))+2)+(x^(4)+1/(x^(4))+2)+(x^(6)+1/(x^(6))+2)+...+(x^(2n)+1/(x^(2n))+2)`
`=(x^(2)+x^(4)+x^(6)+...+x^(2n))`
`+(1/(x^(2))+1/(x^(4))+1/(x^(6))+...+1/(x^(2n)))+underset("n times")((2+3+...))`
`=x^(2)(((x^(2))^(n)-1)/(x^(2)-1))+1/(x^(2))(((1//x^(2))^(n)-1)/((1//x^(2))-1))+2n`
`=x^(2)((x^(2n)-1)/(x^(2)-1))+1/(x^(2n))((1-x^(2n))/(1-x^(2)))+2n`
`=x^(2)((x^(2n)-1)/(x^(2)-1))+1/(x^(2n))((x^(2n)-1)/(x^(2)-1))+2n`
`=((x^(2n)-1)/(x^(2)-1))(x^(2)+1/(x^(2n)))+2n`