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Geometric Mean

If a is the...

Question

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

3 a b c

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2 a b c

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4 a b c

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None of these

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Solution

The correct option is B $2 a b c$
Given that, $2 a = b + c$
and $b, G_{1}, G_{2}, c$ are in G.P.
Therefore, ${G_{1}}^{2} = b G_{2}$ ....(1)
$\Rightarrow {G_{1}}^{3} = b G_{2} G_{1}$
and ${G_{2}}^{2} = c G_{1}$ ....(2)
$\Rightarrow {G_{2}}^{3} = c G_{1} G_{2}$

Multiplying (1) & (2), we get
$G_{1} G_{2} = b c$ ...(3)
Therefore, $G_{1}^{3} + G_{2}^{3} = b G_{2} G_{1} + c G_{2} G_{1} = G_{2} G_{1} (b + c)$
$\Rightarrow G_{1}^{3} + G_{2}^{3} = 2 a b c$ ....[ From (3) ]
Ans: B

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