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Q.
The $(m +n)$ th and $(m-n)$ th terms of a $G.P$. are $p$ and $q$ respectively. Then the $m$ th term of the $G.P$. is
Sequences and Series
Solution:
Let $a$ be the first term and $r$ be the common ratio.
Then, $a_{m +n}=p$ and $a_{m-n}=q$
$\Rightarrow a r^{m+n-1}=p \text { and } a r^{m-n-1}=q$
$\Rightarrow \left(a r^{m+n-1}\right)\left(a r^{m-n-1}\right)=p q$
$\Rightarrow a^{2} r^{2 m-2}=p q$
$\Rightarrow a r^{m-1}=\sqrt{p q}$
$\Rightarrow a_{m}=m \text { th term }=\sqrt{p q}$