Let a and b be the two given numbers.
Then, A.M. = \(\frac{a+b}{2}\) ⇒ \(\frac{a+b}{2}\) ⇒ 2A = a + b ....(i)
As, p and q are G.M’s between a and b, a, p, q, b forms a G.P.
∴ \(\frac{p}{a}\) = \(\frac{q}{p}\) = \(\frac{b}{q}\) ⇒ \(\frac{p^2}{q}\) = a and \(\frac{q^2}{p}\) = b
Now, \(\frac{p^2}{q}\) + \(\frac{q^2}{p}\) = a + b = 2A.