The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Let a and r be the first term and the common ratio of the G.P. respectively.

$\therefore a=1$

$a_{3}=a r^{2}=r^{2}$

$a_{5}=a r^{4}=r^{4}$

$\therefore r^{2}+r^{4}=90$

$\Rightarrow r^{4}+r^{2}-90=0$

$\Rightarrow r^{2}=\frac{-1+\sqrt{1+360}}{2}=\frac{-1 \pm \sqrt{361}}{2}=\frac{-1 \pm 19}{2}=-10$ or 9 $\therefore r=\pm 3$

(Taking real roots)

Thus, the common ratio of the G.P. is ±3.