Question

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

Answer 1

Let the GP be a,ar.$a{r}^2.....$

a = 1,

$T_5+T_5=90$

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

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

$r^2(1+r^2)=90$

Let $r^2=P$ then $\Rightarrow P(1+P)=90$

$\Rightarrow P^2+P-90=0$

$\Rightarrow p^2+10p-9p-90=0$

$\Rightarrow p(p+10)-(p+10)=0$

$\Rightarrow (p-9)(p+10)=0\Rightarrow p=9,-10$

$r^2=9,-10$ so r$=\pm$3 [as the common ratio cannot be a complex No.]