If the difference between the exterior angle of an( n )sided regular polygon and an (n + 1) sided regular polygon is `12^@` find the value of n.

Answer

To solve the problem, we need to find the value of \( n \) such that the difference between the exterior angle of an \( n \)-sided regular polygon and an \( (n + 1) \)-sided regular polygon is \( 12^\circ \).

### Step-by-Step Solution:

1. **Understand the formula for the exterior angle of a polygon**:
The exterior angle of a regular polygon with \( n \) sides is given by the formula:
\[ \text{Exterior Angle} = \frac{360^\circ}{n} \]

2. **Write the exterior angle for \( n \) and \( n + 1 \) sided polygons**:
- For an \( n \)-sided polygon:
\[ \text{Exterior Angle}_n = \frac{360^\circ}{n} \]
- For an \( (n + 1) \)-sided polygon:
\[ \text{Exterior Angle}_{n+1} = \frac{360^\circ}{n + 1} \]

3. **Set up the equation based on the difference**:
According to the problem, the difference between these two exterior angles is \( 12^\circ \):
\[ \frac{360^\circ}{n} - \frac{360^\circ}{n + 1} = 12^\circ \]

4. **Find a common denominator and simplify**:
The common denominator for \( n \) and \( n + 1 \) is \( n(n + 1) \):
\[ \frac{360(n + 1) - 360n}{n(n + 1)} = 12 \]
Simplifying the numerator:
\[ \frac{360}{n(n + 1)} = 12 \]

5. **Cross-multiply to eliminate the fraction**:
\[ 360 = 12n(n + 1) \]

6. **Divide both sides by 12**:
\[ 30 = n(n + 1) \]

7. **Rearrange the equation**:
\[ n^2 + n - 30 = 0 \]

8. **Factor the quadratic equation**:
We need two numbers that multiply to \(-30\) and add to \(1\). These numbers are \(6\) and \(-5\):
\[ (n + 6)(n - 5) = 0 \]

9. **Solve for \( n \)**:
Setting each factor to zero gives:
- \( n + 6 = 0 \) → \( n = -6 \) (not valid since \( n \) must be positive)
- \( n - 5 = 0 \) → \( n = 5 \)

Thus, the value of \( n \) is \( 5 \).

### Final Answer:
\[ n = 5 \]