If one geometric mean G and two arithmetic means `A_1` and `A_2` be inserted between two given quantities, prove that : `G^2=(2A_1-A_2)(2A_2-A_1)`.

If one G.M., G and two A.M's p and q be inserted between two given numbers, prove that G^(2)= (2p-q) (2q-p)

If A_1 and A_2 are two A.M.’s between a and b, prove that : (2A_1-A_2)(2A_2-A_1) =ab .

If A_1 and A_2 are two A.M.’s between a and b, prove that : A_1+A_2 =a+b .

Insert 10 arithmetic means between 2 and 57.

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) = 2abc

If A,A_1,A_2 and A_3 are the areas of the inscribed and escribed circles of a triangle, prove that 1/sqrtA=1/sqrt(A_1)+1/sqrt(A_2)+1/sqrt(A_3)

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.

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 .

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

Two particles A_1 and A_2 of masses m_1 , m_2(m_1>m_2) have the same de-Broglie wavelength then