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If A 1, A 2 ;...

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If $A_{1}, A_{2}; G_{1}, G_{2}$ and $H_{1}, H_{2}$ be AM's, GM's and HM's between two quantities, then the value of $\frac{G_{1} G_{2}}{H_{1} H_{2}}$ is

A

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B

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C

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D

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Solution

The correct option is A
Let the two quantities be a and b. Then $a, A_{1}, A_{2}, b$ are in A.P.
$∴$ $A_{1} - a = b - A_{2} \Rightarrow A_{1} + A_{2} = a + b$ ....(i)
Again $a, G_{1}, G_{2}, b$ are in G.P.
$∴$ $\frac{G_{1}}{a} = \frac{b}{G_{2}} \Rightarrow G_{1} G_{2} = a b$ .....(ii)
Also $a, H_{1}, H_{2}, b$ are in H.P.
$∴ \frac{1}{H_{1}} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H_{2}} \Rightarrow \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{b}$
$\Rightarrow \frac{H_{1} + H_{2}}{H_{1} H_{2}} = \frac{a + b}{a b} = \frac{A_{1} + A_{2}}{G_{1} G_{2}}$ [By (i) and (ii)]
$∴ \frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}} .$

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