Question

Let $T_n$ denote the number of triangles, which can be formed using the vertices of a regular polygon of $n$ side. If $T_{n+1}-T_n=21$, then $n$ equals.

• A. $5$
• B. $7$
• C. $6$
• D. $4$

Answer 1

The correct option is B $7$

A regular polygon of $n$ sides has $n$ vertices, no two of which are collinear. Out of these $n$ points, ${}^n{C}_3$ triangles can be formed.
$△T_n{=}^n{C}_3;T_{n+1}{=}^{n+1}{C}_3$
Given,
$T_{n+1}-T_n=21$
${\Rightarrow }^{n+1}{C}_3{-}^n{C}_3=21$
$\Rightarrow \frac{(n+1)n(n-1)}{3\times 2\times 1}-\frac{n(n-1)(n-2)}{3\times 2\times 1}=21$
$\Rightarrow n(n-1)(n+1-n+2)=126$
$\Rightarrow n(n-1)=42$
$\Rightarrow n(n-1)=7\times 6$
$\Rightarrow n=7$.