Question:

If A1, A2; G1, G2; and H1, H2 are two AMs, GMs and HMs between two quantities, then the value of $\frac{G_{1} G_{2}}{H_{1} H_{2}}$ is

Updated On: Apr 24, 2024

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The Correct Option is A

Explanation:
Let the two quantities be

a

and

b

. Then

a, A_{1}

A_{2}, b

are in AP

∴ A_{1} - a = b - A_{2}

\Rightarrow A_{1} + A_{2} = a + b

Again

a, G_{1}, G_{2}, b

are in GP

∴ \frac{G_{1}}{a} = \frac{b}{G_{2}} \Rightarrow G_{1} G_{2} = a b

Also,

a, H_{1}, H_{2}, b

are in HP.

∴ \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}{a b}

\Rightarrow \frac{H_{1} + H_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{G_{1} G_{2}}

[using Eqs.(i) and

\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}

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