Question

The first term of a GP is $1$. Sum of the third term and fifth term is $90$. Find the common ratio of GP.

Answer 1

Find the common ratio of GP:

Let $r$ be the common ratio and $a$be the first term.

Here it is given that the first term $a=1$

Let $3^{rd}$term be $a{r}^2$and $5^{th}$ term be $a{r}^4$

According to the question,$a{r}^4+a{r}^2=90$

$\begin{array}{cc}\Rightarrow & a{r}^4+a{r}^2=90\\ \Rightarrow & a(r^4+r^2)=90\\ \Rightarrow & r^4+r^2=90\left(\because a=1\right)\\ \Rightarrow & r^4+r^2-90=0\\ \Rightarrow & r^4+10r^2-9r^2-90=0\\ \Rightarrow & r^2(r^2+10)-9(r^2+10)=0\\ \Rightarrow & r=\pm 3\end{array}$

Therefore, the common ratio of GP is $\pm 3$