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 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 a,b,c , are non-zero real numbers such that (1)/(a),(1)/(b),(1)/(c ) are in arithmetic progression and a,b,-2c , are in geometric progression, then which of the following statements (s) is (are) must be true?

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

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 .

If a_1,a_2,……….,a_n are in H.P. then expression a1a2+a2a3+......+a_n-1an

If b , c , B are given and b < c ,prove that cos((A_1-A_2)/2)=(csinB)/b

If G_1 is the mean centre of A_1,B_1,C_1 and G_2 that of A_2,B_2,C_2 then show that vec(A_1A_2)+vec(B_1B_2)+vec(C_1C_2)=3vec(G_1G_2)

Let A_1 , A_2 …..,A_3 be n arithmetic means between a and b. Then the common difference of the AP is

Let a_1,a_2,a_3…………., a_n be positive numbers in G.P. For each n let A_n, G_n, H_n be respectively the arithmetic mean geometric mean and harmonic mean of a_1,a_2,……..,a_n On the basis of above information answer the following question: A_k,G_k,H_k are in (A) A.P. (B) G.P. (C) H.P. (D) none of these