Question

Determine the sum of the series $x(x+y)+x^2(x^2+y^2)+x^3(x^3+y^3)+.......$ to $n$ terms.

Answer 1

$\begin{array}{c}\text{let the sum of the given series be}\\ S_n=x(x+y)+x^2(x^2+y^2)+x^3(x^3+y^3)+\dots \cdots \\ =x^2+xy+x^4+x^2{y}^2+x^6+x^3{y}^3+\dots \cdots \\ =(x^2+x^4+x^6+\dots \dots )+(xy+x^2{y}^2+x^3{y}^3+\dots \cdots )\\ =x^2\cdot (1+x^2+x^4+\dots \text{upto}n\text{terms})+xy(1+xy+x^2{y}^2+\dots \dots \text{upto nterms)}\end{array}$

$\begin{array}{c}=x^2\cdot \left[\frac{1-{\left(x^2\right)}^n}{1-x^2}\right]+xy\cdot \left[\frac{1-(xy{)}^n}{1-xy}\right]\\ =x^2\cdot \left[\frac{1-x^{2n}}{1-x^2}\right]+xy\cdot \left[\frac{1-(xy{)}^n}{1-xy}\right]\end{array}$