Question

If one A.M. A and two G.M.'s p and q be inserted between two given numbers, then $\frac{p^2}{q}+\frac{q^2}{p}=?$

• A. A
• B. $2A$
• C. $A/2$
• D. $A^2$

Answer 1

The correct option is B $2A$

Let the numbers be C & D

Let A be the arithmetic mean

Then $A=\frac{C+D}{2}\Rightarrow 2A=C+D\to \left(1\right)$

Let p,q be the GM between C & D

C,p ,q ,D

So $p^2=cq$ & $q^2=PD$

$\Rightarrow \frac{p^2}{q}=C$ & $\frac{q^2}{p}=D$

$\Rightarrow \frac{p^2}{q}+\frac{q^2}{p}=C+D=2A$ [from eq (1)]

So $\frac{p^2}{q}+\frac{q^2}{p}=2A$