`A_1, A_2,...., A_n` are n A.M’s, and `H_1, H_2,....., H_n` are n H.M’s inserted between a and b . Prove that `(A_1 +A_n)/(H_1+H_n)lt (A^2)/(G^2)`, where A is the arithmetic mean and G is the geometric mean of a and b .

If A_1,A_2,G_1,G_2 and H_1,H_2 are AM’s, GM’s and HM’s between two numbers, then (A_1+A_2)/(H_1+H_2). (H_1H_2)/(G_1G_2) equals _____

If H_(1).H_(2)....,H_(n) are n harmonic means between a and b(!=a), then the value of (H_(1)+a)/(H_(1)-a)+(H_(n)+b)/(H_(n)-b)=

Two AMs. A_1 and A_2 , two GMs. G_1 and G_2 and two HMs. H_1 and H_2 are inserted between any two numbers. Then find the arithmetic mean between H_1 and H_2 in terms of A_1, A_2, G_1, G_2 .

If A1,A2,G1,G2 and H1,H2 are two A.M's , G.Ms and H.M.'s between two numbers,then (A_(1)+A_(2))/(H_(1)+H_(2))*(H_(1)H_(2))/(G_(1)G_(2))

We know that, if a_1, a_2, ..., a_n are in H.P. then 1/a_1,1/a_2,.....,1/a_n are in A.P. and vice versa. If a_1, a_2, ..., a_n are in A.P. with common difference d, then for any b (>0), the numbers b^(a_1),b^(a_2),b^9a_3),........,b^(a_n) are in G.P. with common ratio b^d. If a_1, a_2, ..., a_n are positive and in G.P. with common ration, then for any base b (b> 0), log_b a_1 , log_b a_2,...., log_b a_n are in A.P. with common difference logor.If x, y, z are respectively the pth, qth and the rth terms of an A.P., as well as of a G.P., then x^(z-x),z^(x-y) is equal to

If H_(1),H_(2)…H_(4) be (n-1) harmonic means between a and b, then (H_(1)+a)/(H_(1)-a)+(H_(n-1)+b)/(H_(n-1)-b) is equal to