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)/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.)=...
