<p>Let \(T_n\) be the number of all possible triangles formed by joining vertices of an \(n\)-sided regular polygon. If \(T_{n+1} - T_n = 10\), then the value of \(n\) is</p>
Step-by-Step Solution
Key Concept: Any 3 vertices of an n-sided polygon form a triangle, so T_n = C(n,3). The difference T_{n+1} - T_n represents the triangles gained when adding one new vertex, which equals the number of ways to choose 2 vertices from the existing n vertices: C(n,2).
<p><strong>Step 1:</strong> Express T_n in terms of combinations.</p><p>The number of triangles formed by n vertices of a polygon is T_n = C(n,3) = n(n-1)(n-2)/6</p><p><strong>Step 2:</strong> Find T_{n+1} and compute the difference.</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 = C(n,2)</p><p><strong>Step 3:</strong> Solve the equation.</p><p>n(n-1)/2 = 10</p><p>n(n-1) = 20</p><p>n² - n - 20 = 0</p><p>(n-5)(n+4) = 0</p><p>Since n > 0, we have n = 5</p><p><strong>Verification:</strong> T_5 = C(5,3) = 10, T_6 = C(6,3) = 20, and 20 - 10 = 10 ✓</p><p>∴ Answer: n = 5</p>
Correct Answer: A