Explanation

(a)
Let the two quantities be $a$ and $b$. Then, $a,\ A_{1},\ A_{2},\ b$ are in AP.
$\therefore A_{1} - a = b - A_{2} \Rightarrow A_{1} + A_{2} = a + b\ \ \ldots(i)$
Again $a,\ G_{1},\ G_{2},\ b\$are in GP.
$\therefore\ \frac{G_{1}}{a} = \frac{b}{G_{2}}$
$\Rightarrow \ \ G_{1}G_{2} = ab\ \ \ \$…(ii)
Also, $a,\ H_{1},\ H_{2},\ b\$are in HP.
$\therefore\ \ \frac{1}{H_{1}} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H_{2}}$
$\Rightarrow \ \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{b} + \frac{1}{a}$
$\Rightarrow \ \ \ \frac{H_{1} + H_{2}}{H_{1}H_{2}} = \frac{a + b}{ab}$
$\Rightarrow \ \frac{H_{1} + H_{2}}{H_{1}H_{2}} = \frac{A_{1} + A_{2}}{G_{1}G_{2}}$[using Eqs, (i) and (ii)]
$\Rightarrow \ \frac{G_{1}G_{2}}{H_{1}H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}$