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If one AM ‘A’ and two GM p and q are inserted between two given numbers, then find the value of \(\frac{p^2}{q}\) + \(\frac{q^2}{p}\) in terms of A?

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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.

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