If $H_{1},H_{2}$ are two harmonic means between two positive numbers $a$ and $b\ (a
eq b)$, $A$ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $\frac{H_{2} + H_{1}}{H_{2}H_{1}}$ is

Question

If $H_{1},H_{2}$ are two harmonic means between two positive numbers $a$ and $b\ (a 
eq b)$, $A$ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $\frac{H_{2} + H_{1}}{H_{2}H_{1}}$ is

Dr. Abubacker · Accepted Answer

(d)
Let $a,H_{1},H_{2},b$ are in HP.
$	herefore\ H_{1} = \frac{3ab}{a + 2b}$ , $H_{2} = \frac{3ab}{2a + b}$
Now, $\frac{H_{1} + H_{2}}{H_{1}H_{2}} = \frac{1}{H_{1}} + \frac{1}{H_{2}}$
$= \frac{2a + b}{3ab} + \frac{a + 2b}{3ab} = \frac{a + b}{ab}$ …(i)
Also, $2A = a + b$ …(ii)
and $ab = G^{2}$ …(iii)
From Eqs. (i), (ii) and (iii), we get
$\frac{H_{1} + H_{2}}{H_{1}H_{2}} = \frac{2A}{G^{2}}$

Q. If $H_{1},H_{2}$ are two harmonic means between two positive numbers $a$ and $b\ (a \neq b)$, $A$ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $\frac{H_{2} + H_{1}}{H_{2}H_{1}}$ is

Explanation

Reviewed by Dr. Abubacker

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