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.

Answer 1

Let 'r' be the common ratio for the given G.P.

Here, a = 1 and $a_3+a_5=90$

$\therefore a{r}^2+a{r}^4=90\Rightarrow a(r^2+r^4)=90$

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

which is a quadratic equation in $r^2$

$\therefore r^2=\frac{-1\pm \sqrt{(1{)}^2-4\times (-90)\times 1}}{2\times 1}$

$\Rightarrow r^2=\frac{-1\pm \sqrt{(1{)}^2-4\times (-90)\times 1}}{2\times 1}$

$\Rightarrow r^2=\frac{-1\pm \sqrt{1+360}}{2}=\frac{-1\pm \sqrt{361}}{2}$

$\Rightarrow r^2=\frac{-1\pm 19}{2}$

Either $r^2=\frac{-1+19}{2}$ i.e., $r^2=\frac{18}{2}$

$\Rightarrow r^2=9\Rightarrow r=\pm 3$

or $r^2=\frac{-1-19}{2}$ i.e. $r^2=\frac{-20}{2}$

$\Rightarrow r^2=-10$, which is not possible.

Thus, common ratio of G.P. is $\pm 3$.