Let `A_1, G_1, H_1` denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For `nge2`, let `A_(n-1)` and `H_(n-1)` has arithmetic, geometric and harmonic means as `A_n, G_n, H_n` respectively.
Which of the following statements is correct?

Options

A. `G_1gtG_2gtG_3gt....`
B. `G_1ltG_2ltG_3lt....`
C. `G_1=G_2=G_3=....`
D. `G_1ltG_3ltG_5gt....` and `G_2gtG_4gtG_6gt....`

Answer

To solve the problem, we need to analyze the relationships between the arithmetic mean (A), geometric mean (G), and harmonic mean (H) of two distinct positive numbers. Let's denote these two numbers as \( a \) and \( b \) where \( a < b \).

### Step-by-Step Solution:

1. **Define the Means**:
- The **Arithmetic Mean** \( A_1 \) of \( a \) and \( b \) is given by:
\[ A_1 = \frac{a + b}{2} \]
- The **Geometric Mean** \( G_1 \) is given by:
\[ G_1 = \sqrt{ab} \]
- The **Harmonic Mean** \( H_1 \) is given by:
\[ H_1 = \frac{2ab}{a + b} \]

2. **Recursive Definitions**:
For \( n \geq 2 \), we define:
- \( A_n = \frac{A_{n-1} + H_{n-1}}{2} \)
- \( G_n = \sqrt{A_{n-1} \cdot H_{n-1}} \)
- \( H_n = \frac{2 \cdot A_{n-1} \cdot H_{n-1}}{A_{n-1} + H_{n-1}} \)

3. **Calculate \( A_2 \) and \( H_2 \)**:
- Substitute \( A_1 \) and \( H_1 \) into the formulas for \( A_2 \) and \( H_2 \):
\[ A_2 = \frac{A_1 + H_1}{2} = \frac{\frac{a + b}{2} + \frac{2ab}{a + b}}{2} \]
- Simplifying \( A_2 \):
\[ A_2 = \frac{(a + b)^2 + 4ab}{4(a + b)} = \frac{a^2 + 2ab + b^2 + 4ab}{4(a + b)} = \frac{a^2 + 6ab + b^2}{4(a + b)} \]

4. **Calculate \( G_2 \)**:
- Using the geometric mean formula:
\[ G_2 = \sqrt{A_1 \cdot H_1} = \sqrt{\frac{a + b}{2} \cdot \frac{2ab}{a + b}} = \sqrt{ab} = G_1 \]

5. **Calculate \( H_2 \)**:
- Substitute \( A_1 \) and \( H_1 \) into the formula for \( H_2 \):
\[ H_2 = \frac{2 \cdot A_1 \cdot H_1}{A_1 + H_1} = \frac{2 \cdot \frac{a + b}{2} \cdot \frac{2ab}{a + b}}{\frac{a + b}{2} + \frac{2ab}{a + b}} = \frac{2ab}{\frac{a + b + 4ab/(a + b)}{2}} = H_1 \]

6. **Conclusion**:
- From the calculations, we observe that:
\[ G_n = G_1 \quad \text{for all } n \geq 2 \]
- This implies that the geometric mean remains constant across iterations.

### Correct Statement:
The correct statement is that the geometric means \( G_n \) are equal for all \( n \geq 2 \).