Let `alpha` and `beta` be two positive real numbers. Suppose `A_1, A_2` are two arithmetic means; `G_1 ,G_2` are tow geometrie means and `H_1 H_2` are two harmonic means between `alpha` and `beta`, then

Options

A. `A_(1)H_(2)`
B. `A_(2)H_(1)`
C. `G_(1)G_(2)`
D. None of these

Answer

To solve the problem, we need to find the relationships between the arithmetic means \( A_1, A_2 \), geometric means \( G_1, G_2 \), and harmonic means \( H_1, H_2 \) given two positive real numbers \( \alpha \) and \( \beta \).

### Step 1: Find the Arithmetic Means \( A_1 \) and \( A_2 \)

The two arithmetic means \( A_1 \) and \( A_2 \) between \( \alpha \) and \( \beta \) can be expressed as:
\[ A_1 = \alpha + \frac{1}{3}(\beta - \alpha) = \frac{2\alpha + \beta}{3} \]
\[ A_2 = \beta - \frac{1}{3}(\beta - \alpha) = \frac{\alpha + 2\beta}{3} \]

### Step 2: Calculate \( A_1 + A_2 \)

Now, we can calculate \( A_1 + A_2 \):
\[ A_1 + A_2 = \frac{2\alpha + \beta}{3} + \frac{\alpha + 2\beta}{3} = \frac{3\alpha + 3\beta}{3} = \alpha + \beta \]
This gives us our first equation:
\[ A_1 + A_2 = \alpha + \beta \quad \text{(Equation 1)} \]

### Step 3: Find the Geometric Means \( G_1 \) and \( G_2 \)

The geometric means \( G_1 \) and \( G_2 \) can be expressed as:
\[ G_1 = \sqrt{\alpha \cdot \beta} \]
\[ G_2 = \sqrt{\alpha \cdot \beta} \]
Thus, we have:
\[ G_1 \cdot G_2 = \sqrt{\alpha \cdot \beta} \cdot \sqrt{\alpha \cdot \beta} = \alpha \cdot \beta \]
This gives us our second equation:
\[ G_1 \cdot G_2 = \alpha \cdot \beta \quad \text{(Equation 2)} \]

### Step 4: Find the Harmonic Means \( H_1 \) and \( H_2 \)

The harmonic means \( H_1 \) and \( H_2 \) can be expressed as:
\[ \frac{1}{H_1} = \frac{1}{2} \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \quad \Rightarrow \quad H_1 = \frac{2\alpha\beta}{\alpha + \beta} \]
\[ \frac{1}{H_2} = \frac{1}{2} \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \quad \Rightarrow \quad H_2 = \frac{2\alpha\beta}{\alpha + \beta} \]

### Step 5: Calculate \( H_1 + H_2 \) and \( H_1 \cdot H_2 \)

Calculating \( H_1 + H_2 \):
\[ H_1 + H_2 = \frac{2\alpha\beta}{\alpha + \beta} + \frac{2\alpha\beta}{\alpha + \beta} = \frac{4\alpha\beta}{\alpha + \beta} \]

Calculating \( H_1 \cdot H_2 \):
\[ H_1 \cdot H_2 = \left( \frac{2\alpha\beta}{\alpha + \beta} \right) \cdot \left( \frac{2\alpha\beta}{\alpha + \beta} \right) = \frac{4\alpha^2\beta^2}{(\alpha + \beta)^2} \]

### Step 6: Establish the Relationship

Now we have:
- \( A_1 + A_2 = \alpha + \beta \)
- \( G_1 \cdot G_2 = \alpha \cdot \beta \)
- \( H_1 + H_2 = \frac{4\alpha\beta}{\alpha + \beta} \)
- \( H_1 \cdot H_2 = \frac{4\alpha^2\beta^2}{(\alpha + \beta)^2} \)

### Conclusion

From the above calculations, we can conclude that:
- The relationship between the arithmetic, geometric, and harmonic means can be expressed as:
\[ \frac{A_1 + A_2}{H_1 + H_2} = \frac{G_1 \cdot G_2}{H_1 \cdot H_2} \]