<p>The interior angles of a polygon are in AP. If the smallest angle be 120° and the common difference be 5, then the number of sides is</p>
Step-by-Step Solution
Key Concept: The sum of interior angles of an n-sided polygon is (n-2)×180°. If the angles form an AP with first term a and common difference d, their sum can be expressed as n×a + n(n-1)d/2. Equating these two expressions gives us an equation to solve for n.
<p><strong>Step 1:</strong> Identify given information. The interior angles are in AP with smallest angle a = 120° and common difference d = 5°.</p><p><strong>Step 2:</strong> Write the sum of interior angles using AP formula. For n angles in AP: Sum = n/2[2a + (n-1)d] = n/2[2(120) + (n-1)(5)] = n/2[240 + 5n - 5] = n/2[235 + 5n]</p><p><strong>Step 3:</strong> Write the sum of interior angles of a polygon. For an n-sided polygon: Sum = (n-2)×180°</p><p><strong>Step 4:</strong> Equate the two expressions: n/2[235 + 5n] = (n-2)×180</p><p><strong>Step 5:</strong> Simplify: n[235 + 5n] = 360(n-2)</p><p>235n + 5n² = 360n - 720</p><p>5n² + 235n - 360n + 720 = 0</p><p>5n² - 125n + 720 = 0</p><p><strong>Step 6:</strong> Divide by 5: n² - 25n + 144 = 0</p><p><strong>Step 7:</strong> Factor: (n - 9)(n - 16) = 0</p><p>So n = 9 or n = 16</p><p><strong>Step 8:</strong> Check validity. For n = 16: largest angle = 120 + 15(5) = 195° > 180°, which is impossible for an interior angle of a polygon. For n = 9: largest angle = 120 + 8(5) = 160° < 180°, which is valid.</p><p><strong>∴ Answer:</strong> c</p>
Correct Answer: c