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Expansion of (a ± b ± c)²

If one geomet...

Question

If one geometric mean $G$ and two arithmetic means $A_{1}$ and $A_{2}$ are inserted between two given quantities, then $(2 A_{1} - A_{2}) (2 A_{2} - A_{1})$

A

G

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B

G^{2}

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C

G^{3}

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D

2 G

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Solution

The correct option is A $G^{2}$
Let $a$ and $b$ be two quantities. Then,
$G = \sqrt{a b}, A_{1} = \frac{2 a + b}{3}, A_{2} = \frac{a + 2 b}{3}$
$∴ 2 A_{1} - A_{2} = a$
$2 A_{1} - A_{2} = b$
$\Rightarrow (2 A_{1} - A_{2}) (2 A_{1} - A_{2}) = a b = G^{2}$

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