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 $T_{n+1}-T_n=21$, then $n$ equals

• A. $5$
• B. $7$
• C. $6$
• D. $4$

Answer 1

The correct option is B

$7$

Explanation for the correct option:

Find the value of $n$.

In the question, it is given that $T_n$ is the number of triangles which can be formed by using the vertices of a regular polygon of $n$ sides.

So, the value of $T_n$ is ${}^{n}C_3$.

Now, find the value of $n$ as follows:

Since, $T_{n+1}-T_n=21$.

$$\begin{gathered} {}^{n+1}C_3-{}^{n}C_3=21 \\ \Rightarrow {}^{n+1}C_3-{}^{n}C_3=21 \\ \Rightarrow \frac{\left(n+1\right)n(n-1)}{3·2·1}-\frac{n(n-1)(n-2)}{3·2·1}=21 \\ \Rightarrow \frac{n(n-1)\left[n+1-n+2\right]}{6}=21 \\ \Rightarrow \frac{n(n-1)\left[3\right]}{6}=21 \\ \Rightarrow \frac{n(n-1)}{2}=21 \\ \Rightarrow n(n-1)=42 \end{gathered}$$

Which is possible only when $n=7$.

Therefore, the value of $n$ is $7$.

Hence, option $B$ is the correct option.