If $G_{1}$ and $G_{2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_{1}}^{2}}{G_{2}} + \frac{{G_{2}}^{2}}{G_{1}}$ is

$\frac{A}{2}$

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$A$

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$2 A$

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None of these

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Solution

The correct option is C
$2 A$

Explanation for the correct option:
Finding the value of $\frac{{G_{1}}^{2}}{G_{2}} + \frac{{G_{2}}^{2}}{G_{1}}$ :
Let, $a$ and $b$ are the two numbers, then the arithmetic mean
$\begin{array}{rcl} A & = & \frac{a + b}{2} \\ \Rightarrow & a + b = 2 A . . . (1) \end{array}$
Geometric mean inserted between two number,
$a, G_{1}, G_{2}, b$ are in GP, then
$\begin{array}{rcl} {G_{1}}^{2} & = & a G_{2} [∵ b^{2} = a c] \\ \Rightarrow & \frac{{G_{1}}^{2}}{G_{2}} = a . . . (2) \\ {G_{2}}^{2} & = & b G_{1} \\ \Rightarrow & \frac{{G_{2}}^{2}}{G_{1}} = b . . . (3) \end{array}$
Add equation $(2)$ and $(3)$ , we get
$\begin{array}{rcl} \frac{{G_{1}}^{2}}{G_{2}} + \frac{{G_{2}}^{2}}{G_{1}} & = & a + b \\ = & 2 A [from 1] \end{array}$
Hence, the correct option is C.

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If one A.M., A and two geometric means $G_{1}$ and $G_{2}$ inserted between any two positive numbers, show that $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}} = 2 A$ .

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and

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such that

\frac{G_{3} - G_{2}}{G_{2} - G_{1}} = 2

, then

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If G1 and G2 are two geometric means and A the arithmetic mean inserted between two numbers, then the value of G12G2+G22G1 is

If $G_{1}$ and $G_{2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_{1}}^{2}}{G_{2}} + \frac{{G_{2}}^{2}}{G_{1}}$ is