Question

The sum of $2n$ terms of a G.P. whose first term is $a$ and common ratio $r$ is equal to the sum of $n$ of a G.P. whose first term is $b$ and common ratio $r^2$. Prove that $b$ is equal to the sum of the first two terms of the first series.

Answer 1

Series 1.
First term $=a$
Common ratio $=r$
Terms $a,ar,a{r}^2,....,a{r}^{2n-1}$

Series 2.
First term $=b$
Common ratio $=r^2$
Given that
$\frac{a(r^{2n}-1)}{r-1}=\frac{b\left[\right(r^2{)}^n-1]}{r^2-1}$
$\Rightarrow a=\frac{b}{r+1}$
$\Rightarrow b=a(r+1)=a+ar$