Question

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 if it is positive.

Answer 1

Let $a$ and $r$ be the first term and the common ratio of the G.P. respectively.
$\therefore a=1a_3={ar}^2=r^2{a}_5={ar}^4=r^4$
Given $a_3+a_5=90$
$\therefore r^2+r^4=90\Rightarrow r^4+r^2=90\Rightarrow r^4+r^2-90=0\Rightarrow r^2=\frac{-1+\sqrt{1+360}}{2}=\frac{-1\pm \sqrt{361}}{2}=\frac{-1\pm 19}{2}=-10or9\therefore r=\pm 3\left(\text{Taking real roots}\right)$