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,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 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 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)

If H_(1) , H_(2) are two harmonic means between two positive numbers a and b , (a != b) , A and G are the arithmetic and geometric means between a and b , then (H_(2) + H_(1))/(H_(2) H_(1)) is

If H_(1),H_(2) are two harmonic means between two positive numbers a and b (aneb) , A and G are the arithmetic and geometric means between a and b, then (H_(2)+H_(1))/(H_(2)H_(1)) is

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)=((2a+b)(a+2b))/(9a b)

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)=((2a+b)(a+2b))/(9a b)

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)=((2a+b)(a+2b))/(9a b)

Let h_(n) and g_(n) be harmonic and geometric sequences respectively. If h_(1) = g_(1) = 1/2 and h_(10) = g_(10) = (1)/(1024) then