Question

$x(x+y)+x^2(x^2+y^2)+x^3(x^3+y^3)+....$ to $n$ terms.

Answer 1

Let $S$ be the sum of the series
Then, $S=x(x+y)+x^2(x^2+y^2)+x^3(x^3+y^3)+....$ to $n$ terms.
Given series can be written as sum of two series whose sum is denoted by $S_1,S_2$
Let's take $S_1=x^2+x^4+x^6+............x^{2n}$
Clearly it is a G.P. with common ratio $r=x^2$
$\therefore S_1=\frac{x^2(x^{2n}-1)}{x^2-1}$
$S_2=xy+(xy{)}^2+(xy{)}^3+(xy{)}^4+............(xy{)}^n$
Clearly it is a G.P. with common ratio $r=xy$
$S_2=\frac{xy\left[\right(xy{)}^n-1]}{xy-1}$
$S=S_1+S_2$
$\Rightarrow S=\frac{x^2(x^{2n}-1)}{x^2-1}+\frac{xy\left[\right(xy{)}^n-1]}{xy-1}$