If between two numbers p and q, there are inserted two arithmetic means `A_(1)`, `A_(2)` , two geometric means `G_(1)`, `G_(2)` and two harmonic means `H_(1)`, `H_(2)` then 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 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))

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

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 G_(1) and G_(2) are two geometric means and A is the arithmetic mean inserted two numbers,G_(1)^(2) then the value of (G_(1)^(2))/(G_(2))+(G_(2)^(2))/(G_(1)) is:

If one A.M.A and tow geometric means G_(1) and G_(2) inserted between any two positive numbers,show that (G_(1)^(1))/(G_(2))+(G_(2)^(2))/(G_(1))=2A

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

If one geometric mean G and two arithmetic means A_(1) and A_(2) be inserted between two given quantities,prove that G^(2)=(2A_(1)-A_(2))(2A_(2)-A_(1))

If A_(1),A_(2),G_(1),G_(2) and H_(1),H_(2) be two AMs,GMs and HMs between two quantities then the value of (G_(1)G_(2))/(H_(1)H_(2)) is