If a,b are two numbers, by their harmonic mean is meant c such that `1/a,1/c,1/b` are in A.P. If a,b are positive and A,G,H denote respectively the arithmetic, geometric and harmonic ,means of a,b, show that A,G,H form a G.P.

If a,b,c are in A.P. then :

If 1/a,1/b,1/c are in A.P and a,b -2c, are in G.P where a,b,c are non-zero then

If A and G are the A.M. and G.M. respectively between any two distinct positive numbers a and b then show that A > G.

If a, b, c are in G.P. and x, y are arithmetic mean of a, b and b, c respectively, then 1/x +1/y is equal to :

Let a_1, a_2 ,........ be positive real numbers in geometric progression. For each n, let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a_1,a_2..........a_n . Find an expression ,for the geometric mean of G_1,G_2,........G_n in terms of A_1, A_2,........ ,A_n, H_1, H_2,........,H_n .

Let n in N ,n > 25. Let A ,G ,H deonote te arithmetic mean, geometric man, and harmonic mean of 25 and ndot The least value of n for which A ,G ,H in {25 , 26 .....n} is

If A, G and H are the arithmetic mean, geometric mean and harmonic mean between unequal positive integers. Then, the equation Ax^2 -|G|x- H=0 has

If a, b, c are in A.P, b, c, d are in G.P. and 1/c,1/d,1/e are in A.P. prove that a, c, e are in G.P.

If G_1 is the first of n geometric means between a and b, show that : G_1^(n+1) =a^n b .

If a(1/b +1/c) , b(1/c +1/a), c(1/a + 1/b) are in A.P. Prove that a,b,c are in A.P.