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`.

Determine the number n in a geometric progression {a_n} , if a_1= 3, a_n= 96 and S_n= 189 .

If A, G and H are the arithmetic mean, geometric mean and harmonic mean between unequal positive integers. Then, the equation Ax^2 -|G|x- H=0 has

Let n in N ,n > 25. Let A ,G ,H deonote te arithmetic mean, geometric man, and harmonic mean of 25 and ndot The least value of n for which A ,G ,H in {25 , 26 .....n} is

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

The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G^2=27. Find two numbers.

If a_1,a_2,a_3,....a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1+a_2+....+a_(n-1)+3a_n is

Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively a. A_(1)gtA_(2)gtA_(3)gt"......." b. A_(1)ltA_(2)ltA_(3)lt"......." c. A_(1)gtA_(3)gtA_(5)gt"......."" and " A_(2)ltA_(4)ltA_(6)lt"......." d. A_(1)ltA_(3)ltA_(5)lt"......."" and " A_(2)gtA_(4)gtA_(6)gt"......."

If G_1 is the first of n geometric means between a and b, show that : G_1^(n+1) =a^n b .

Let A_(1),G_(1),H_(1) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For nge2 , let A_(n-1),G_(n-1)" and "H_(n-1) has arithmetic,geometric and harmonic means as A_(n),G_(n),H_(n) respectively. Which of the following statement is correct?

Let A_(1),G_(1),H_(1) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For nge2 , let A_(n-1),G_(n-1)" and "H_(n-1) has arithmetic,geometric and harmonic means as A_(n),G_(n),H_(n) respectively. Which of the following statement is correct? a. H_(1)gtH_(2)gtH_(3)gt"......." b. H_(1)ltH_(2)ltH_(3)lt"......." c. H_(1)gtH_(3)gtH_(5)gt"......." " and " H_(2)ltH_(4)ltH_(6)lt"......." d. H_(1)ltH_(3)ltH_(5)lt"......."" and " H_(2)gtH_(4)gtH_(6)gt"......."