The A.M., G.M. and H.M. between any two unequal positive numbers satisfy the relations (a) (G.M.)2  = (A.M.) x (G.M.)             (b) A.M. > G.M. > H.M.

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)/2 x 2ab/(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.) = (G.M.)/(H.M.) 

Since, A.M > G.M.
we have G.M. > H.M.
Combining the two, We have
 A.M. > G.M. > H.M.