The difference between an exterior angle of `(n-1)` sided regular polygon and an exterior angle of `(n+2)` sided regular polygon is `6^(@)`. Find the value of n

Answer

To solve the problem, we need to find the value of \( n \) given that the difference between the exterior angle of an \( (n-1) \)-sided regular polygon and that of an \( (n+2) \)-sided regular polygon is \( 6^\circ \).

### Step-by-Step Solution:

1. **Understand the formula for exterior angles**:
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 angles for both polygons**:
- For the \( (n-1) \)-sided polygon:
\[ \text{Exterior angle of } (n-1) = \frac{360^\circ}{n-1} \]
- For the \( (n+2) \)-sided polygon:
\[ \text{Exterior angle of } (n+2) = \frac{360^\circ}{n+2} \]

3. **Set up the equation based on the difference**:
According to the problem, the difference between these two exterior angles is \( 6^\circ \):
\[ \frac{360^\circ}{n-1} - \frac{360^\circ}{n+2} = 6^\circ \]

4. **Simplify the equation**:
To simplify, we can factor out \( 360^\circ \):
\[ 360^\circ \left( \frac{1}{n-1} - \frac{1}{n+2} \right) = 6^\circ \]
Dividing both sides by \( 360^\circ \):
\[ \frac{1}{n-1} - \frac{1}{n+2} = \frac{1}{60} \]

5. **Combine the fractions**:
The left-hand side can be combined:
\[ \frac{(n+2) - (n-1)}{(n-1)(n+2)} = \frac{3}{(n-1)(n+2)} \]
Thus, we have:
\[ \frac{3}{(n-1)(n+2)} = \frac{1}{60} \]

6. **Cross-multiply to solve for \( n \)**:
Cross-multiplying gives:
\[ 3 \cdot 60 = (n-1)(n+2) \]
Simplifying this:
\[ 180 = (n-1)(n+2) \]

7. **Expand the right-hand side**:
Expanding gives:
\[ 180 = n^2 + 2n - n - 2 \]
Simplifying further:
\[ 180 = n^2 + n - 2 \]

8. **Rearranging the equation**:
Rearranging gives us a standard quadratic equation:
\[ n^2 + n - 182 = 0 \]

9. **Factoring the quadratic equation**:
We need to find two numbers that multiply to \(-182\) and add to \(1\). The factors are \(14\) and \(-13\):
\[ (n - 13)(n + 14) = 0 \]

10. **Finding the values of \( n \)**:
Setting each factor to zero gives:
\[ n - 13 = 0 \quad \Rightarrow \quad n = 13 \]
\[ n + 14 = 0 \quad \Rightarrow \quad n = -14 \quad (\text{not valid since } n \text{ must be positive}) \]

Thus, the value of \( n \) is \( 13 \).