If `A_1,A_2,G_1,G_2` and `H_1,H_2` are AM’s, GM’s and HM’s between two numbers, then `(A_1+A_2)/(H_1+H_2). (H_1H_2)/(G_1G_2)` equals _____

Answer

To solve the problem, we need to find the value of the expression \((A_1 + A_2)/(H_1 + H_2) \cdot (H_1 H_2)/(G_1 G_2)\) given that \(A_1, A_2\) are arithmetic means, \(G_1, G_2\) are geometric means, and \(H_1, H_2\) are harmonic means between two numbers \(a\) and \(b\).

### Step 1: Define the Arithmetic Means
Since \(A_1\) and \(A_2\) are the arithmetic means between \(a\) and \(b\), we can express this as:
\[ A_1 + A_2 = a + b \]
This is because the arithmetic means \(A_1\) and \(A_2\) are the two numbers that complete the arithmetic progression.

**Hint:** Remember that the sum of the arithmetic means is equal to the sum of the two numbers they are between.

### Step 2: Define the Geometric Means
For the geometric means \(G_1\) and \(G_2\):
\[ G_1 G_2 = ab \]
This follows from the property of geometric means, where the product of the means is equal to the product of the two numbers.

**Hint:** The geometric mean of two numbers is the square root of their product.

### Step 3: Define the Harmonic Means
For the harmonic means \(H_1\) and \(H_2\):
\[ \frac{2}{H_1 + H_2} = \frac{1}{a} + \frac{1}{b} \]
This implies:
\[ H_1 + H_2 = \frac{2ab}{a + b} \]

**Hint:** The harmonic mean is defined in terms of the reciprocals of the numbers.

### Step 4: Substitute into the Expression
Now, we substitute the values we found into the expression:
\[ \frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} \]
Substituting \(A_1 + A_2 = a + b\) and \(H_1 + H_2 = \frac{2ab}{a + b}\):
\[ \frac{a + b}{\frac{2ab}{a + b}} \cdot \frac{H_1 H_2}{ab} \]

### Step 5: Simplify the First Part
The first part simplifies as follows:
\[ \frac{(a + b)^2}{2ab} \]

**Hint:** When dividing by a fraction, multiply by its reciprocal.

### Step 6: Find \(H_1 H_2\)
Using the harmonic mean property:
\[ H_1 H_2 = \frac{2ab}{H_1 + H_2} = \frac{2ab}{\frac{2ab}{a + b}} = a + b \]

### Step 7: Substitute \(H_1 H_2\) into the Expression
Now, substituting \(H_1 H_2 = a + b\) into the expression:
\[ \frac{(a + b)^2}{2ab} \cdot \frac{a + b}{ab} = \frac{(a + b)^3}{2ab^2} \]

### Step 8: Final Simplification
The final expression simplifies to:
\[ \frac{(a + b)^3}{2ab^2} \]

### Conclusion
Thus, the value of the expression \((A_1 + A_2)/(H_1 + H_2) \cdot (H_1 H_2)/(G_1 G_2)\) equals:
\[ \frac{(a + b)^3}{2ab^2} \]