Let $a_{1}, a_{2}, \ldots, a_{10}$ be a G.P. If $\frac{a_{3}}{a_{1}}=25,$ then $\frac{a_{9}}{a_{5}}$ equals :
Let $a_{1}, a_{2}, \ldots, a_{10}$ be a G.P. If $\frac{a_{3}}{a_{1}}=25,$ then $\frac{a_{9}}{a_{5}}$ equals :
$5^{4}$
$4\left(5^{2}\right)$
$5^{3}$
$2\left(5^{2}\right)$
Solution
Let $a_{1}=a, a_{2}=a r, a_{3}=a r^{2} \ldots a_{10}=a r^{9}$
where $r=$ common ratio of given G.P.
Given, $\frac{a_{3}}{a_{1}}=25$
$\Rightarrow \frac{a r^{2}}{a}=25$
$\Rightarrow r=\pm 5$
Now, $\frac{a_{9}}{a_{5}}=\frac{a r^{8}}{a r^{4}}=r^{4}=(\pm 5)^{4}=5^{4}$