Question

Let $A_1$ , $A_2$ be two arithmetic means , $G_1$ , $G_2$ be two geometric means and $H_1$ , $H_2$ be two harmonic means between two positive numbers a and b . The value of $\frac{G_1{G}_2}{H_1{H}_2}.\frac{H_1+H_2}{A_1+A_2}$ is

• A. 1/2
• B. 1
• C. 3/2
• D. 2

Answer 1

The correct option is B 1

Let the two numbers be, $a,b$

$a,A_1,A_2,b$ are in A.P

$\frac{a+b}{2}=\frac{A_1+A_2}{2}$

$\therefore a+b=A_1+A_2$.........(1)

$a,G_1,G_2,b$ are in G.P

$\sqrt{ab}=\sqrt{G_1{G}_2}$

$\therefore ab=G_1{G}_2$.........(2)

$a,H_1,H_2,B$ are in H.P

$\frac{\frac{1}{a}+\frac{1}{b}}{2}=\frac{\frac{1}{H_1}+\frac{1}{H_2}}{2}$

$\frac{a+b}{ab}=\frac{H_1+H_2}{H_1{H}_2}$........(3)

Now,

$\frac{G_1{G}_2}{H_1{H}_2}\cdot \frac{H_1+H_2}{A_1+A_2}$

$\frac{G_1{G}_2}{A_1+A_2}\cdot \frac{H_1+H_2}{H_1{H}_2}$

from (1), (2) and (3), we have,

$\frac{ab}{a+b}\cdot \frac{a+b}{ab}$

$=1$