Surfaces of a thin equi-convex glass lens have radius of curvature R. Paraxial rays are incident on it . If the final image is formed after n internal reflections, calculate distance of this image from pole of the lens. Refractive index of glass is `mu`.

Explanation

To solve the problem, we will follow a systematic approach to calculate the distance of the final image from the pole of a thin equi-convex glass lens after n internal reflections.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Radius of curvature of the lens, \( R \)
- Refractive index of glass, \( \mu \)
- Number of internal reflections, \( n \)

2. **Refraction at the First Surface:**
- For the first refraction, we use the lens maker's formula:
\[ \frac{\mu}{V_i} - \frac{1}{\infty} = \frac{\mu - 1}{R} \]
- Since rays are coming from infinity, \( \frac{1}{\infty} = 0 \):
\[ \frac{\mu}{V_i} = \frac{\mu - 1}{R} \]
- Rearranging gives:
\[ V_i = \frac{\mu R}{\mu - 1} \]

3. **Reflection at the First Surface:**
- Using the lens formula for the first reflection:
\[ \frac{1}{V_1} + \frac{1}{V_i} = \frac{1}{F} \]
- The focal length \( F \) for a thin lens can be approximated as:
\[ F = \frac{R}{2(\mu - 1)} \]
- Thus, substituting \( V_i \) into the lens formula:
\[ \frac{1}{V_1} + \frac{\mu - 1}{\mu R} = \frac{2(\mu - 1)}{R} \]
- Rearranging gives:
\[ \frac{1}{V_1} = \frac{2(\mu - 1)}{R} - \frac{\mu - 1}{\mu R} \]
- Simplifying further leads to:
\[ V_1 = -\frac{3(\mu - 1)}{\mu R} \]

4. **Reflection at Subsequent Surfaces:**
- For the second reflection, we follow a similar process:
\[ \frac{1}{V_2} + \frac{1}{V_1} = \frac{1}{F} \]
- Continuing this process for \( n \) reflections, we find:
\[ \frac{1}{V_n} = -\frac{(2n + 1)(\mu - 1)}{\mu R} \]

5. **Final Refraction:**
- After \( n \) reflections, we consider the final refraction:
\[ \frac{\mu}{V_f} - \frac{1}{V_n} = \frac{\mu - 1}{R} \]
- Rearranging gives:
\[ V_f = \frac{R}{2\mu n + \mu - 1} \]

### Final Result:
The distance of the final image from the pole of the lens is:
\[ V_f = \frac{R}{2\mu n + \mu - 1} \]