The nth division of main scale coincides with (n +1) th division of vernier scale. Given one main division is equal to 'a' units. Find the least count of the vernier.

Explanation

To find the least count of the vernier scale given that the nth division of the main scale coincides with the (n + 1)th division of the vernier scale, and that one main scale division (msd) is equal to 'a' units, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that the nth division of the main scale (MS) coincides with the (n + 1)th division of the vernier scale (VS).
- We are given that 1 main scale division (1 msd) is equal to 'a' units.

2. **Write the Relationship**:
- From the problem, we can express the relationship as:
\[ n \text{ msd} = (n + 1) \text{ v sd} \]
- This means that the measurement represented by n divisions of the main scale is equal to the measurement represented by (n + 1) divisions of the vernier scale.

3. **Express 1 Vernier Scale Division (v sd)**:
- We can rearrange the equation to find the value of 1 vernier scale division:
\[ 1 \text{ v sd} = \frac{n \text{ msd}}{n + 1} \]
- Substituting the value of 1 msd (which is 'a' units):
\[ 1 \text{ v sd} = \frac{n \cdot a}{n + 1} \]

4. **Use the Formula for Least Count**:
- The formula for the least count (LC) of a vernier scale is given by:
\[ \text{Least Count} = 1 \text{ msd} - 1 \text{ v sd} \]
- We already know that 1 msd = 'a' units and we have calculated 1 v sd.

5. **Substitute the Values**:
- Now, substituting the values into the least count formula:
\[ \text{Least Count} = a - \frac{n \cdot a}{n + 1} \]

6. **Simplify the Expression**:
- To simplify this expression:
\[ \text{Least Count} = a \left(1 - \frac{n}{n + 1}\right) \]
- This simplifies to:
\[ \text{Least Count} = a \left(\frac{(n + 1) - n}{n + 1}\right) = a \left(\frac{1}{n + 1}\right) \]

7. **Final Result**:
- Therefore, the least count of the vernier scale is:
\[ \text{Least Count} = \frac{a}{n + 1} \text{ units} \]