Arithmetic, Geometric, and Harmonic Mean

The arithmetic mean of two numbers is the sum of the numbers divided by 2, while the geometric mean is the square root of the product of the numbers. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers. That is, if a and b are two distinct, positive numbers, then the arithmetic, geometric, and harmonic means are given by:

Arithmetic Mean: (a + b)/2
Geometric Mean: √ab
Harmonic Mean: 2ab/(a + b)

Solution

Given that the arithmetic and geometric means of two distinct, positive numbers are A and G, respectively. Let a and b be the two numbers such that a ≠ b. Therefore,

A = (a + b)/2
G = √ab

We are to find the harmonic mean of the two numbers, which is given by:

2ab/(a + b)

Substituting the values of A and G in the above expression, we get:

2ab/[(a + b)/2 + √ab]

Multiplying both the numerator and denominator by 2, we get:

4ab/(a + b + 2√ab)

Now, we can substitute the value of G in the denominator:

4ab/(a + b + 2G)

Substituting the value of A in the numerator:

4ab/(2A + 2G)

Dividing both the numerator and denominator by 2G, we get:

2ab/G(A + G)

Substituting the value of G/A from the expression for the arithmetic mean, we get:

G/A = √ab/[(a + b)/2] = 2√ab/(a + b)

Substituting this value in the above expression, we get:

2ab/[G(A + G)] = G2/A

Therefore, the harmonic mean of two distinct, positive numbers whose arithmetic and geometric means are A and G, respectively, is given by G2/A. Hence, option C is the correct answer.