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 $G.P.$
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 $ $a=1$ $a_{3}=a r^{2}=r^{2} \quad a_{5}=a r^{4}=r^{4}$
$\therefore r^{2}+r^{4}=90$
$\Rightarrow r^{4}+r^{2}-90=0$
$\Rightarrow r^{2}=\frac{-1+\sqrt{1+360}}{2}=\frac{-1 \pm \sqrt{361}}{2}=-10$ or $9$
$\therefore r=\pm 3$ [ Taking real roots ]
Thus, the common ratio of the $G.P.$ is $±3$ .
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