Question

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

• A. $5$
• B. $7$
• C. $6$
• D. $4$

Answer 1

Answer: (B), (C)

$6$

Explanation for the correct option:

Finding the value of $n$

Given that

$T_n$ the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides.

We know that

$T_n={}^n{C}_r$ Here $r=3$

Also, given

$T_{n+1}-T_n=21$

$$\begin{gathered} \Rightarrow {}^{n+1}{C}_3+{}^n{C}_3=21 \\ \Rightarrow \frac{\left(n+1\right)\left(n\right)(n-1)}{3\times 2\times 1}+\frac{n\left(n-1\right)\left(n-2\right)}{3\times 2\times 1}=21 \\ \Rightarrow \frac{n\left(n-1\right)}{2}=21 \\ \Rightarrow n^2-n=42 \\ \Rightarrow n^2-n-42=0 \\ \Rightarrow n^2-7n+6n-42=0 \\ \Rightarrow n\left(n-7\right)-6\left(n-7\right)=0 \\ \Rightarrow \left(n-7\right)\left(n-6\right)=0 \\ \Rightarrow n=7or6 \end{gathered}$$
Therefore, $n=7$ or $6$

Hence, option (B) and (C) is the correct answer.