Let `A_1 , G_1, H_1`denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For `n >2,`let `A_(n-1),G_(n-1)` and `H_(n-1)` has arithmetic, geometric and harmonic means as `A_n, G_N, H_N,` respectively.

Options

A. `A_(1)gtA_(2)gtA_(3)gt"......."`
B. `A_(1)ltA_(2)ltA_(3)lt"......."`
C. `A_(1)gtA_(3)gtA_(5)gt"......."" and " A_(2)ltA_(4)ltA_(6)lt"......."`
D. `A_(1)ltA_(3)ltA_(5)lt"......."" and " A_(2)gtA_(4)gtA_(6)gt"......."`

Explanation

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, and how these means behave as we iterate through them.

### Step-by-Step Solution:

1. **Define the Means for Two Distinct Positive Numbers:**
Let \( a \) and \( b \) be two distinct positive numbers such that \( a < b \).
- The **Arithmetic Mean** \( A_1 \) 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. **Establish the Relationship Between the Means:**
It is known that for any two distinct positive numbers:
\[ A_1 > G_1 > H_1 \]
This means the arithmetic mean is always greater than the geometric mean, which in turn is greater than the harmonic mean.

3. **Define the Means for Subsequent Iterations:**
For \( n > 2 \), we define:
- \( A_n \) as the arithmetic mean of \( A_{n-1} \) and \( H_{n-1} \):
\[ A_n = \frac{A_{n-1} + H_{n-1}}{2} \]
- \( G_n \) as the geometric mean of \( A_{n-1} \) and \( H_{n-1} \):
\[ G_n = \sqrt{A_{n-1} H_{n-1}} \]
- \( H_n \) as the harmonic mean of \( A_{n-1} \) and \( H_{n-1} \):
\[ H_n = \frac{2 A_{n-1} H_{n-1}}{A_{n-1} + H_{n-1}} \]

4. **Analyze the Behavior of the Means:**
Since \( A_{n-1} > H_{n-1} \), we can conclude that:
- \( A_n \) will always be greater than \( H_n \) because the arithmetic mean of two numbers is always greater than the harmonic mean of the same two numbers.
- Similarly, \( G_n \) will also fall between \( A_n \) and \( H_n \).

5. **Induction on the Means:**
By induction, we can show that:
- \( A_1 > A_2 > A_3 > \ldots \)
- \( H_1 < H_2 < H_3 < \ldots \)

This means that as \( n \) increases, the arithmetic mean decreases and the harmonic mean increases.

6. **Conclusion:**
From the analysis, we conclude that:
- The sequence of arithmetic means \( A_n \) is strictly decreasing.
- The sequence of harmonic means \( H_n \) is strictly increasing.

Therefore, the correct option is:
\[ A_1 > A_2 > A_3 > \ldots \]

### Final Answer:
The correct option is the first one: \( A_1 > A_2 > A_3 > \ldots \)