If the sum of an infinite GP

Sum of infinite terms :

$\frac{a}{1-r}=15$             .....(i)

Series formed by square of terms:

$a^{2}, a^{2} r^{2}, a^{2} r^{4}, a^{2} r^{6} \ldots . .$

Sum $=\frac{\mathrm{a}^{2}}{1-\mathrm{r}^{2}}=150$

$\Rightarrow \frac{\mathrm{a}}{1-\mathrm{r}} \cdot \frac{\mathrm{a}}{1+\mathrm{r}}=150 \Rightarrow 15 \cdot \frac{\mathrm{a}}{1+\mathrm{r}}=150$

$\Rightarrow \frac{\mathrm{a}}{1+\mathrm{r}}=10$           ......(ii)

by (i) and (ii) $\mathrm{a}=12 ; \mathrm{r}=\frac{1}{5}$

Now series : $a r^{2}, a r^{4}, a r^{6}$

Sum $=\frac{a r^{2}}{1-r^{2}}=\frac{12 \cdot(1 / 25)}{1-1 / 25}=\frac{1}{2}$