Question

Let \( T_n \) denote the number of triangles which can be formed by using the vertices of a regular polygon of \( n \) sides. If \( T_{n+1} - T_n = 36 \), then \( n \) is equal to:

• A. \( 2 \)
• B. \( 5 \)
• C. \( 6 \)
• D. \( 8 \)
• E. \( 9 \)

Hint:

$\binom{n}{2}$ represents the number of handshakes among $n$ people or the number of lines between $n$ points. If you know that 36 lines can be formed from 9 points, you can solve this instantly.

Answer 1

Answer: (E)

Understanding the Concept: The number of triangles formed by \( n \) vertices of a regular polygon is the number of ways to choose 3 vertices out of \( n \), which is \( T_n = \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \).

Step 1: Use the property of combinations.
Recall the identity \( \binom{n+1}{r} - \binom{n}{r} = \binom{n}{r-1} \). In this problem: \[ T_{n+1} - T_n = \binom{n+1}{3} - \binom{n}{3} = \binom{n}{2} \]

Step 2: Set up the equation.
Given \( T_{n+1} - T_n = 36 \): \[ \binom{n}{2} = 36 \] \[ \frac{n(n-1)}{2} = 36 \]

Step 3: Solve for \( n \).
\[ n(n-1) = 72 \] Since \( 9 \times 8 = 72 \), we have \( n = 9 \).