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

The correct option is E 9

Since no 3 points in a regular polygon are collinear, a triangle can be formed by selecting any 3 points.

$\therefore$ Number of triangle that can be formed by using vertices of a regular polygon of $n$ sides ${=}^n{C}_3$

$\therefore T_{n+1}-T_n{=}^{n+1}{C}_3{-}^n{C}_3=\frac{(n+1)!}{(n-2)!3!}-\frac{n!}{(n-3)!3!}=36$

$⟹\frac{(n+1)\cdot n!}{(n-2)!3!}-\frac{(n-2)\cdot n!}{(n-2)!3!}=\frac{n!}{2(n-2)!}=36$

$⟹n(n-1)=72⟹n^2-n-72=0⟹(n-9)(n+8)=0$

$n=9,n=-8$

Since $n$, cannot be negative, $n=9$ Option E is the correct answer