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Q.

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 $b$ , then values $\frac{G_{1} G_{2}}{H_{1} H_{2}} \cdot \frac{H_{1} + H_{2}}{A_{1} + A_{2}}$ is

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a

$\frac{1}{2}$

b

$\frac{3}{2}$

c

1

d

2

answer is C.

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Detailed Solution

We have $A_{1} + A_{2} = a + b$ ,
$G_{1} G_{2} = a b$ and $\frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{b} .$
$\begin{aligned} ∴ \frac{G_{1} G_{2}}{H_{1} H_{2}} \cdot \frac{H_{1} + H_{2}}{A_{1} + A_{2}} = \frac{G_{1} G_{2}}{A_{1} + A_{2}} (\frac{1}{H_{1}} + \frac{1}{H_{2}}) \\ = \frac{a b}{a + b} (\frac{1}{a} + \frac{1}{b}) = 1 . \end{aligned}$

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Let A1, A2 be two arithmetic means, G1, G2 be two geometric means, and H1, H2 be two harmonic means between two positive numbers a and b, then values G1G2H1H2⋅H1+H2A1+A2is