Find the sum to n terms of the sequence : (i) `(x+1/x)^(2)+ (x^(2)+1/x^(2))^(2)+ (x^(3)+1/x^(3))^(2)` ... to n terms(ii) `(x+y)+ (x^(2)+xy+y^(2))+(x^(3)+x^(2)y+xy^(2)+y^(3))`, ... to n terms

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

1.	Find the sum to n terms of the sequence : (i) `(x+1/x)^(2)+ (x^(2)+1/x^(2))^(2)+ (x^(3)+1/x^(3))^(2)` ... to n terms(ii) `(x+y)+ (x^(2)+xy+y^(2))+(x^(3)+x^(2)y+xy^(2)+y^(3))`, ... to n terms
Answer» Correct Answer - (i) `((x^(2n)-1)/(x^(2)-1))(x^(2)+1/x^(2n))+2n` (ii) `1/((x-y)).{x^(2)((x^(n)-1)/(x-1))-y^(2)((y^(n)-1)/(y-1))}` (i) Given sum `=(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)` `=(a^(2)+x^(4)+x^(6)+...+x^(2n))+(1/x^(2)+1/x^(4)+1/x^(6)+...+1/x^(2n))+(2+2+..."n imes")` `=(x^(2)[(x^(2))^(n)-1])/((x^(2)-1))+(1/x^(2).{1-(1/x^(2))^(n)})/((1-1/x^(2)))+2n=(x^(2)(x^(2n)-1))/((x^(2)-1))+((x^(2n)-1))/(x^(2n) (x^(2)-1))+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`. (ii) Multiplying and dividing each term by `(x-y)`, we get the given sum as `1/((x-y)).{(x^(2)-y^(2))+(x^(3)-y^(3))+(x^(4)-y^(4))+..."to n terms"}` `=1/((x-y)).{(x^(2)+x^(3)+x^(4)+..."to n terms")-(y^(2)+y^(3)+y^(4)+..."to n terms")}` `=1/((x-y)).{x^(2)((x^(n)-1)/(x-1))-y^(2) ((y^(n)-1)/(y-1))}`.