If x^2, b^2, y^2 are in AP, a, x, b and b, y, c both are in GP, prove that a, b, c are in AP.

Let a and b be two unequal positive numbers. Then,

A.M. =(a+b)/2 G.M =(ab)

and, H.M. = 2ab/(a+b)
To Prove the first part, we have

(A.M.) x(G.M.) =(a+b)/2x2ab/(a+b)

= ab
= (ab)²

To prove the second part, let us consider that

A.M. – G.M. =(a+b)/2-(ab)

= {(a + b - 2(ab)}/2

= 1/2(a-b)²

which is always positive as square of any number either positive or negative always yield positive number

Hence, A.M. > G.M.
Again,
(A.M.) x(H.M.) = (G.M.) x(G.M.)
or, (A.M.)/ (G.M.)=...