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Arithmetic Mean

If 'a' be t...

Question

If $^{'} a^{'}$ be the $A M$ between $b$ and $c$ and $G M^{'} s$ are $G_{1}$ and $G_{2}$ , then $G_{1}^{3} + G_{2}^{3}$ is equal to

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 C $2 a b c$
Since, $b, a$ and $c$ are in AP.
Thus $2 a = b + c$ ..... (i)
and $b, G_{1}, G_{2}$ and $c$ are in GP.
$G_{1} = b r, G_{2} = b r^{2}$ and $c = b r^{3}$ ,
where $r$ be the common ratio of GP.
Now, $G_{1}^{3} + G_{2}^{3} = (b r)^{3} + (b r^{2})^{3}$
$= b^{3} r^{3} + b^{3} r^{6}$
$= b^{3} (\frac{c}{b}) + b^{3} {(\frac{c}{b})}^{2}$
$= b^{2} c + b c^{2} = b c (b + c) = 2 a b c$ ..[From Eq. (i)]
Therefore, $G_{1}^{3} + G_{2}^{3} = 2 a b c$ .

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