If a is the A.M. of b and c and the two geometric means are `G_(1)` and `G_(2)`, then prove that `G_(1)^(3)+G_(2)^(3)`

If a be one A.M and G_1 and G_2 be then geometric means between b and c then G_1^3+G_2^3=

If a is the arithmetic mean and g is the geometric mean of two numbers, then

If G is the geometric mean of xa n dy then prove that 1/(G^2-x^2)+1/(G^2-y^2)=1/(G^2)

If H_(1) , H_(2) are two harmonic means between two positive numbers a and b , (a != b) , A and G are the arithmetic and geometric means between a and b , then (H_(2) + H_(1))/(H_(2) H_(1)) 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)

If a, b and c are in G.P and x and y, respectively , be arithmetic means between a,b and b,c then prove that a/x+c/y=2 and 1/x+1/y=2/b

if m is the AM of two distinct real number l and n and G_1,G_2 and G_3 are three geometric means between l and n , then (G_1^4 + G_2^4 +G_3^4 ) equals to

If m is the A.M. of two distinct real numbers l and n""(""l ,""n"">""1) and G1, G2 and G3 are three geometric means between l and n, then G1 4+2G2 4+G3 4 equals, (1) 4l^2 mn (2) 4l^m^2 mn (3) 4l m n^2 (4) 4l^2m^2n^2