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)`

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)) .

In direct proportion a_1/b_1 = a_2/b_2

If 2, h_(1), h_(2),………,h_(20)6 are in harmonic progression and 2, a_(1),a_(2),……..,a_(20), 6 are in arithmetic progression, then the value of a_(3)h_(18) is equal to

If a, b and c are positive numbers in arithmetic progression and a^(2), b^(2) and c^(2) are in geometric progression, then a^(3), b^(3) and c^(3) are in (A) arithmetic progression. (B) geometric progression. (C) harmonic progression.

Let (1+x^2^2)^2(1+x)^n = sum _(k=0)^(n+4)a_k x^k. If a_1,a_2,a_3 are in rithmetic progression find n.

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

If a_1,a_2,a_3 …. are in harmonic progression with a_1=5 and a_20=25 . Then , the least positive integer n for which a_n lt 0 , 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 each n, let A_n G_n, H_n, be respectively the arithmetic mean, geometric mean & 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.