In a $G . P$ if the ${(m + n)}^{t h}$ term be $p$ and ${(m - n)}^{t h}$ term be $q$ , then prove that its $m^{t h}$ term is $\sqrt{p q}$ .

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Solution

$T_{m + n} = a r^{m + n - 1} = p$ .
$T_{m - n} = a r^{m - n - 1} = q$ and $T_{m} = a r^{m - 1}$
Multiplying $a^{2} r^{2 m - 2} = p q$
$∴ T_{m} = a r^{m - 1} = \sqrt{p q}$ .

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