First slide

Means - A.M/G.M/H.M

Question

if A is the arithmetic mean and G₁, G₂ be two geometric means between any two numbers, then $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}}$ is equal to

Moderate

A

2A
B

A
C

3A
D

None of these

Solution

Let A be the arithmetic mean between two numbers a and b, Again, let G₁,andG₂, be the two geometric mean between two numbers a and b.
$∴ A = \frac{a + b}{2} and G_{1} = ar, G_{2} = {ar}^{2}$
where $r = {(\frac{b}{a})}^{\frac{1}{2 + 1}} = {(\frac{b}{a})}^{1 / 3}$ ……(i)
$∴ \frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}} = \frac{a^{2} r^{2}}{{ar}^{2}} + \frac{a^{2} r^{4}}{ar}$
$= a + {ar}^{3} = a + a (\frac{b}{a}) [∵ from Eq. (i)]$
$= a + b = 2 A [∵ A = \frac{a + b}{2}]$

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