Let `a ,b` be positive real numbers. If `a A_1, A_2, b` be are in arithmetic progression `a ,G_1, G_2, b` are in geometric progression, and `a ,H_1, H_2, b` are in harmonic progression, show that `(G_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)`

Let a,b be positive real numbers.If aA_(1),A_(2),b be are in arithmetic progression a,G_(1),G_(2),b are in geometric progression,and a,H_(1),H_(2),b are in harmonic progression,show that (G_(1)G_(2))/(H_(1)H_(2))=(A_(1)+A_(2))/(H_(1)+H_(2))

Let alpha and beta be two positive real numbers. Suppose A_1, A_2 are two arithmetic means; G_1 ,G_2 are two geometric means and H_1 H_2 are two harmonic means between alpha and beta , then

Let alpha and beta be two positive real numbers. Suppose A_1, A_2 are two arithmetic means; G_1 ,G_2 are tow geometrie means and H_1 H_2 are two harmonic means between alpha and beta , then

Let alpha and beta be two positive real numbers. Suppose A_1, A_2 are two arithmetic means; G_1 ,G_2 are tow geometrie means and H_1 H_2 are two harmonic means between alpha and beta , then

Let alpha and beta be two positive real numbers. Suppose A_1, A_2 are two arithmetic means; G_1 ,G_2 are tow geometrie means and H_1 H_2 are two harmonic means between alpha and beta , then

Lef a, b be two distinct positive numbers and A and G their arithmetic and geometric means respectively. If a, G_(1),G_(2) , b are in G.P. and a, H_(1),H_(2) , b are in H.P., prove that (G_(1)G_(2))/(H_(1)H_(2))lt (A^(2))/(G^(2)) .

If a,A_(1),A_(2),b are in A.P., a,G_(1),G_(2),b are in G.P. and a, H_(1),H_(2),b are in H.P., then the value of G_(1)G_(2)(H_(1)+H_(2))-H_(1)H_(2)(A_(1)+A_(2)) is

Let a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, 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 a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, 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 a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, 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)