Question

Let $T_n$ denotes 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 $n$ is equal to

• A. $5$
• B. $7$
• C. $6$
• D. $4$

Answer 1

The correct option is B $7$

Number of triangle that can be formed using polygon of $n$ sided $={}^n{C}_3$

According to question,

${}^{n+1}{C}_3-{}^n{C}_3=21$

$\frac{(n+1)\left(n\right)(n-1)}{6}-\frac{n(n-1)(n-2)}{6}=21$

$\Rightarrow \frac{3\left(n\right)(n-1)}{6}=21$

$\Rightarrow \left(n\right)(n-1)=42$

Solving this equation we get $n=-6,7$

But $n$ can't be negative

Hence, $n=7$.