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 $
$\therefore n=9$