Question

A geometric progression has the first term $a$, common ratio $r$ and sum to infinity $S$. A second geometric progression has first term $a$, common ratio $2r$ and sum to infinity $3S$. Find the value of $r$.

Answer 1

First G.P: $a,r$

Second G.P: $a,2r$

the sum of infinite G.P=$\frac{a}{1-r}$

$\begin{array}{c}s=\frac{a}{1-r}..........\left(1\right)\\ 3s=\frac{a}{1-2r}..........\left(2\right)\\ \frac{eq\left(2\right)}{eq\left(1\right)}=\frac{\frac{a}{1-2r}}{\frac{a}{1-r}}=\frac{3s}{s}\\ 3=\frac{1-r}{1-2r}\\ r=\frac{2}{5}\end{array}$