<p>The interior angles of a polygon are in arithmetic progression. The smallest angle is \(120^\circ\), and the common difference is \(5^\circ\). Find the number of sides of the polygon.</p>
Step-by-Step Solution
Key Concept: The sum of interior angles of an n-sided polygon equals (n-2)×180°. Since angles form an AP, use the arithmetic series sum formula S = n/2(2a + (n-1)d) and equate to (n-2)×180°.
<p><strong>Step 1:</strong> Set up the sum of interior angles. For an n-sided polygon with angles in AP: a = 120°, d = 5°</p><p>Sum of angles = n/2[2(120) + (n-1)(5)] = (n-2)×180</p><p><strong>Step 2:</strong> Simplify the left side: n/2[240 + 5n - 5] = n/2[235 + 5n] = n(235 + 5n)/2</p><p><strong>Step 3:</strong> Set equal to polygon angle sum: n(235 + 5n)/2 = 180(n-2)</p><p>n(235 + 5n) = 360(n-2)</p><p>235n + 5n² = 360n - 720</p><p>5n² - 125n + 720 = 0</p><p>n² - 25n + 144 = 0</p><p><strong>Step 4:</strong> Factor: (n - 9)(n - 16) = 0, so n = 9 or n = 16</p><p><strong>Step 5:</strong> Check validity. For n = 16: largest angle = 120 + 15(5) = 195° > 180° (invalid for a polygon)</p><p>For n = 9: largest angle = 120 + 8(5) = 160° < 180° ✓</p><p>∴ Answer: <strong>9</strong></p>
Correct Answer: 9