The difference between an exterior angle of n -... - JAMB Mathematics 2023 Question

Let's denote the exterior angle of an \(n\)-sided regular polygon as \(E_n\). The exterior angle of an \(n\)-sided regular polygon is given by:

\[ E_n = \frac{360^\circ}{n} \]

Now, according to the given information:

\[ E_{n-1} - E_{n+2} = 6^\circ \]

Substitute the expressions for \(E_{n-1}\) and \(E_{n+2}\) into the equation:

\[ \frac{360^\circ}{n-1} - \frac{360^\circ}{n+2} = 6^\circ \]

To solve this equation, we can find a common denominator:

\[ \frac{360(n+2) - 360(n-1)}{(n-1)(n+2)} = 6 \]

Simplify the numerator:

\[ \frac{360n + 720 - 360n + 360}{(n-1)(n+2)} = 6 \]

Combine like terms:

\[ \frac{1080}{(n-1)(n+2)} = 6 \]

Now, cross-multiply and simplify:

\[ 1080 = 6(n-1)(n+2) \]

Divide both sides by 6:

\[ 180 = (n-1)(n+2) \]

Expand the right side:

\[ 180 = n^2 + 2n - n - 2 \]

Combine like terms:

\[ 180 = n^2 + n - 2 \]

Rewrite the equation in standard form:

\[ n^2 + n - 182 = 0 \]

Factor the quadratic expression:

\[ (n - 13)(n + 14) = 0 \]

This gives two possible solutions:

\[ n - 13 = 0 \quad \text{or} \quad n + 14 = 0 \]

\[ n = 13 \quad \text{or} \quad n = -14 \]

Since the number of sides (\(n\)) cannot be negative, we discard the solution \(n = -14\). Therefore, the correct value of \(n\) is:

\[ n = 13 \]