First slide

Geometric progression

Question

If G1 and G2 are two geometric means and A is the arithmetic mean inserted between two positive numbers then the value of $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}}$ is

Moderate

A

A/2
B

A
C

2A
D

none of these

Solution

$G_{1} = a r, G_{2} = a r^{2}$ where $r = {(\frac{b}{a})}^{1 / 3}$ .
$\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}} = \frac{G_{1}^{3} + G_{2}^{3}}{G_{1} G_{2}} = \frac{a^{3} r^{3} + a^{3} r^{6}}{a^{2} r^{3}}$
$= a + a r^{3} = a + b = 2 A$

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