Question

Sum the series : `x(x+y)+x^2(x^2+y^2)+x^3(x^3+y^3+ ton` terms.

Answer 1

`x(x+y)+x^2(x^2+y^2)+....n` terms
`=>x^2+xy+x^4+x^2y^2+x^6+x^3y^3+...n` terms
`=>(x^2+x^4+x^6+...n ` terms `)` +`(xy+(xy)^2+(xy)^3+...n` terms`)`
Now, we have `2` G.P. with first term and common ratio `x^2` and `xy`.
Sum of a G.P. can be given as,
`S_n = (a(r^n-1))/(r-1)`
`:. S_n = (x^2(x^(2n) - 1))/(x^2-1)+(xy(xy^n - 1))/(xy-1)`