If `a_1,a_2,...,a_n` be n positive real numbers in ascending order of magnitude, prove that `a_1<(a_1^2+a_2^2+....+a_n^2)/(a_1+a_2+...+a_n)

If a_1,a_2, .....,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)+2a_n is a. a_(n-1)+2a_n b. (n+1)c^(1//n) c. 2n c^(1//n) d. (n)(2c)^(1//n)

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)+2a_n is

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

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

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

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

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

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

.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.