<p>If \(T_n\) = Number of triangles formed using the vertices of an \(n\)-sided regular polygon and \(T_{n+1} - T_n = 21\), then \(n\) equals:</p>
Step-by-Step Solution
Key Concept: Triangles from n vertices require choosing any 3 non-collinear points. For a regular polygon, all sets of 3 vertices form valid triangles since no three vertices are collinear. The difference T_{n+1} - T_n gives the number of new triangles when adding one vertex.
<p><strong>Step 1:</strong> For an n-sided regular polygon, any 3 vertices form a triangle (no three vertices are collinear).</p><p>T_n = C(n,3) = n(n-1)(n-2)/6</p><p><strong>Step 2:</strong> Calculate T_{n+1} - T_n:</p><p>T_{n+1} = C(n+1,3) = (n+1)n(n-1)/6</p><p>T_{n+1} - T_n = (n+1)n(n-1)/6 - n(n-1)(n-2)/6</p><p>= n(n-1)/6 [(n+1) - (n-2)]</p><p>= n(n-1)/6 × 3</p><p>= n(n-1)/2</p><p><strong>Step 3:</strong> Set equal to 21:</p><p>n(n-1)/2 = 21</p><p>n(n-1) = 42</p><p>n² - n - 42 = 0</p><p>(n-7)(n+6) = 0</p><p>n = 7 (taking positive value)</p><p><strong>Verification:</strong> T_7 = C(7,3) = 35; T_8 = C(8,3) = 56; T_8 - T_7 = 21 ✓</p><p>∴ Answer: n = 7</p>
Correct Answer: C