Question:
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
(a) 0
(b) pq
(c) $\sqrt{p q}$
(d) $\frac{1}{2}(p+q)$
Solution:
(c) $\sqrt{p q}$
Here, $a_{(m+n)}=p$
$\Rightarrow a r^{(m+n-1)}=p \quad \ldots \ldots(\mathrm{i})$
Also, $\mathbf{a}_{(m-n)}=q$
$\Rightarrow a r^{(m-n-1)}=q \quad \ldots \ldots$ (ii)
Mutliplying (i) and (ii):
$\Rightarrow a r^{(m+n-1)} a r^{(m-n-1)}=p q$
$\Rightarrow a^{2} r^{(2 m-2)}=p q$
$\Rightarrow\left(a r^{(m-1)}\right)^{2}=p q$
$\Rightarrow a r^{(m-1)}=\sqrt{p q}$
$\Rightarrow a_{m}=\sqrt{p q}$
Thus, the $m^{\text {th }}$ term is $\sqrt{p q}$.
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