If `H_n=1+1/2+1/3+.....+1/n, AA n in N,` then `H_1+H_2+H_3+......+H_n=`

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 .

.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 .

.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 .

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 and 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 .

.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 H_n=1+1/2+...+1/ndot , then the value of S_n=1+3/2+5/3+...+(99)/(50) is a. H_(50)+50 b. 100-H_(50) c. 49+H_(50) d. H_(50)+100

If H_n=1+1/2+...+1/ndot , then the value of S_n=1+3/2+5/3+...+(99)/(50) is a. H_(50)+50 b. 100-H_(50) c. 49+H_(50) d. H_(50)+100

If x_(1) x_(2) ...... x_(n)&(1)/(h_(1)) (1)/(h_(2)) ......(1)/(h_(n)) are two such that x_(3)=h_(2)=8&x_(8)=h_(7)=20 then x_(5)h_(10) is

Show that H_1 + H_2 +…....+H_(n) = (n+1)H_(n) - n. where H_(n) = 1 + (1)/(2) + (1)/(3) +….+ (1)/(n) AA n in N.

If H_1,H_2,H_3,.......H_n be n harmonic means between a and b then (H_1+a)/(H_1-a)+(H_n+b)/(H_n-b)=