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Geometric Mean

If A.M. of b ...

Question

If A.M. of $b$ and $c$ is $a$ and the two geometric means between them are $G_{1}$ and $G_{2}$ , then the value of $G_{1}^{3} + G_{2}^{3}$ is

a b c

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2 a b c

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3 a b c

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4 a b c

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Solution

The correct option is B $2 a b c$
Given : A.M. of $b$ and $c$ is $a$ , so
$2 a = b + c$

Now, $b, G_{1}, G_{2}, c$ are in G.P., so
$b r^{3} = c \Rightarrow r = {(\frac{c}{b})}^{1 / 3}$

Therefore,
$G_{1} = b r, G_{2} = b r^{2} \Rightarrow G_{1}^{3} + G_{2}^{3} = b^{3} r^{3} (1 + r^{3}) \Rightarrow G_{1}^{3} + G_{2}^{3} = b^{2} c [1 + \frac{c}{b}] \Rightarrow G_{1}^{3} + G_{2}^{3} = b c (b + c) ∴ G_{1}^{3} + G_{2}^{3} = 2 a b c$

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