If one geometric mean G and two arithmetic means p,q be inserted between two given numbers, then prove that, `G^2=(2p - q) (2q - p)`.

Explanation

Let the two quantities be a and b, then
`G^(2)=ab" " "……(i)"`
Again, `a,p,q,b` are in AP.
`therefore p-a=q-p=b-q`
`implies a=2p-q " " "……..(ii)"`
From Eqs. (i) and (ii), we get
`G^(2)=(2p-q)(2q-p)`