First slide

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

Question

If two geometric means g1 and g2 and one arithmetic mean A be inserted between two numbers, then $\frac{g_{1}^{2}}{g_{2}} + \frac{g_{2}^{2}}{g_{1}} =$

Moderate

A

4A
B

3A
C

2A
D

A

Solution

Let the two numbers be a and b.
$∴ A = \frac{a + b}{2} or 2 A = a + b$ (1)
Again, a, g₁, g₂, b are in G.P.
$∴ \frac{g_{1}}{a} = \frac{g_{2}}{g_{1}} = \frac{b}{g_{2}}$
Now, $\frac{g_{1}}{a} \equiv \frac{g_{2}}{g_{1}} \Rightarrow \frac{g_{1}^{2}}{g_{2}} \equiv a$
and $\frac{g_{2}}{g_{1}} = \frac{b}{g_{2}} \Rightarrow \frac{g_{2}^{2}}{g_{1}} = b$
$∴ a + b = \frac{g_{1}^{2}}{g_{2}} + \frac{g_{2}^{2}}{g_{1}}$ (2)
$∴$ From Eqs (1) and (2), we get
$2 A = \frac{g_{1}^{2}}{g_{2}} + \frac{g_{2}^{2}}{g_{1}}$

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