If $A_{1}, A_{2}$ are two AM's $G_{1}, G_{2}$ are two GM's and $H_{1}, H_{2}$ are two HM's between two number, then $\frac{A_{1} + A_{2}}{H_{1} + H_{2}}$ is equal to

\frac{H_{1} H_{2}}{G_{1} G_{2}}

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\frac{G_{1} G_{2}}{H_{1} H_{2}}

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\frac{H_{1} H_{2}}{A_{1} A_{2}}

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\frac{G_{1} G_{2}}{A_{1} A_{2}}

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Solution

The correct option is A $\frac{G_{1} G_{2}}{H_{1} H_{2}}$
$G = \sqrt{A H}$
here $A = \frac{A_{1} + A_{2}}{2}, G = \sqrt{G_{1} G_{2}}, H = \frac{2 H_{1} H_{2}}{H_{1} + H_{2}}$
$\Rightarrow A H = \frac{A_{1} + A_{2}}{H_{1} + H_{2}} \times H_{1} H_{2} = G^{2} = G_{1} G_{2}$
$\Rightarrow \frac{A_{1} + A_{2}}{H_{1} + H_{2}} = \frac{G_{1} G_{2}}{H_{1} H_{2}}$

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