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

Answer 1

Answer: (E)

Given, $T_{n+1}-T_n=36$
$\therefore {}^{n+1}{C}_3-{}^n{C}_3=36$
$\Rightarrow \frac{(n+1)n(n-1)}{3\cdot 2\cdot 1}-\frac{n(n-1)(n-2)}{3\cdot 2\cdot 1}=36$
$\Rightarrow \frac{n(n-1)}{6}[n+1-n+2]=36$
$\Rightarrow \frac{n(n-1)\cdot 3}{6}=36$
$\Rightarrow n(n-1)=72$
$\Rightarrow n(n-1)=9\times 8$
$\therefore n=9$