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

• (A) $\frac{A_1+A_2}{H_1+H_2}$
• (B) $\frac{A_1-A_2}{H_1+H_2}$
• (C) $\frac{A_1+A_2}{H_1-H_2}$
• (D) $\frac{A_1-A_2}{H_1-H_2}$

Answer 1

The Correct Option is A

Explanation:
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$$\Rightarrow A_1+A_2=a+b$Again $a,G_1,G_2,b$ are in GP$\therefore \frac{G_1}{a}=\frac{b}{G_2}\Rightarrow G_1{G}_2=ab$Also, $a,H_1,H_2,b$ are in HP.$\therefore \frac{1}{H_1}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H_2}$$\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 $\frac{G_1{G}_2}{H_1{H}_2}=\frac{A_1+A_2}{H_1+H_2}$