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Geometric Mean

If a be one...

Question

If $a$ be one A.M and $G_{1}$ and $G_{2}$ be the G.Ms between $b$ and $c$ then $G_{1}^{3} + G_{2}^{3} =$

A

a b c

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B

2 a b c

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C

3 a b c

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D

4 a b c

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Solution

The correct option is B $2 a b c$
Lets consider $b = p^{3}, c = q^{3}$
$G 1 = p^{2} q$
$G 2 = p q^{2}$
$(G 1)^{3} + (G 2)^{3} = (p q)^{3} [p^{3} + q^{3}]$
$= b c (b + c)$
$a = (b + c) / 2$
$2 a = b + c$
So the expression for $(G 1)^{3} + (G 2)^{3} = 2 a b c$
Option B

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