For each natural number k. Let `C_(k)` denotes the circle with radius k centimetres and centre at origin. On the circle `C_(k)` a particle moves k centrimetres in the counter-clockwise direction . After completing its motion on `C_(k)` the particle moves to `C_(k+1)` in the radial direction . The motion of the particle continue in this manner. The particle starts at ( 1,0). If the particle crosses the positive direction of the X -axis for the first time on the circle `C_(n)`, then `n ="__________________"`.

Answer

To solve the problem, we will analyze the motion of the particle step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Position**:
The particle starts at the point (1, 0) on the circle \(C_1\) which has a radius of 1 cm.

2. **Motion on Circle \(C_k\)**:
For each circle \(C_k\), the particle moves \(k\) centimeters in the counter-clockwise direction. The angle \(\theta\) in radians that the particle moves can be calculated using the formula:
\[ \theta = \frac{\text{arc length}}{\text{radius}} = \frac{k}{k} = 1 \text{ radian} \]

3. **Position After Moving on Circle \(C_1\)**:
After moving 1 cm on circle \(C_1\), the particle moves 1 radian counter-clockwise from (1, 0). The new position can be calculated using the parametric equations of a circle:
\[ x = r \cos(\theta) = 1 \cos(1) \quad \text{and} \quad y = r \sin(\theta) = 1 \sin(1) \]

4. **Moving to Circle \(C_2\)**:
After completing the motion on \(C_1\), the particle moves radially outward to circle \(C_2\) (radius 2 cm). The new position is:
\[ x = 2 \cos(1) \quad \text{and} \quad y = 2 \sin(1) \]

5. **Motion on Circle \(C_2\)**:
The particle then moves 2 cm on circle \(C_2\). The angle moved is:
\[ \theta = \frac{2}{2} = 1 \text{ radian} \]
The new position after moving on \(C_2\) is:
\[ x = 2 \cos(1 + 1) = 2 \cos(2) \quad \text{and} \quad y = 2 \sin(1 + 1) = 2 \sin(2) \]

6. **Continuing the Process**:
This process continues for each \(k\):
- On circle \(C_k\), the particle moves \(k\) cm, which corresponds to moving 1 radian.
- After moving on \(C_k\), the particle moves to circle \(C_{k+1}\).

7. **Finding When the Particle Crosses the Positive X-axis**:
The particle crosses the positive x-axis when the y-coordinate is 0 and the x-coordinate is positive. This occurs when:
\[ \theta = 2\pi n \quad \text{for some integer } n \]
The total angle moved after \(n\) circles is:
\[ \text{Total angle} = 1 + 1 + 1 + \ldots + 1 = n \text{ radians} \]
We need:
\[ n = 2\pi \implies n \approx 6.28 \]
Since \(n\) must be a natural number, we take \(n = 7\).

### Final Answer:
Thus, the particle crosses the positive direction of the X-axis for the first time on the circle \(C_7\).

**Answer**: \(n = 7\)