The difference between the exterior angles of two regular polygons, having the sides equal to (n-1) and (n+1) is `9^@.` Find the value of n.

Answer

To solve the problem step by step, we will find the value of \( n \) given that the difference between the exterior angles of two regular polygons with sides equal to \( (n-1) \) and \( (n+1) \) is \( 9^\circ \).

### Step 1: Write the formula for the exterior angle of a polygon
The exterior angle \( E \) of a regular polygon with \( s \) sides is given by the formula:
\[ E = \frac{360^\circ}{s} \]

### Step 2: Calculate the exterior angles for both polygons
For the first polygon with \( (n-1) \) sides:
\[ E_1 = \frac{360^\circ}{n-1} \]
For the second polygon with \( (n+1) \) sides:
\[ E_2 = \frac{360^\circ}{n+1} \]

### Step 3: Set up the equation based on the difference of the exterior angles
According to the problem, the difference between these two exterior angles is \( 9^\circ \):
\[ E_1 - E_2 = 9^\circ \]
Substituting the values of \( E_1 \) and \( E_2 \):
\[ \frac{360^\circ}{n-1} - \frac{360^\circ}{n+1} = 9^\circ \]

### Step 4: Simplify the equation
To simplify the left-hand side, find a common denominator:
\[ \frac{360^\circ(n+1) - 360^\circ(n-1)}{(n-1)(n+1)} = 9^\circ \]
This simplifies to:
\[ \frac{360^\circ(n + 1 - n + 1)}{(n-1)(n+1)} = 9^\circ \]
\[ \frac{720^\circ}{(n-1)(n+1)} = 9^\circ \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 720^\circ = 9^\circ \cdot (n-1)(n+1) \]

### Step 6: Simplify the equation
Dividing both sides by \( 9^\circ \):
\[ 80 = (n-1)(n+1) \]
Using the identity \( (a-b)(a+b) = a^2 - b^2 \):
\[ 80 = n^2 - 1 \]

### Step 7: Solve for \( n^2 \)
Adding \( 1 \) to both sides:
\[ n^2 = 81 \]

### Step 8: Find \( n \)
Taking the square root of both sides:
\[ n = \pm 9 \]
Since \( n \) must be a positive integer (as it represents the number of sides of a polygon), we have:
\[ n = 9 \]

### Final Answer
The value of \( n \) is \( 9 \).