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. $6$
• C. $7$
• D. None of these

Answer 1

The correct option is C $7$

We see that $T_n={{}^{n}C}_3$

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

$\Rightarrow \frac{(n+1)\left(n\right)(n-1)}{\mathrm{1.2.3}}-\frac{n(n-1)(n-2)}{\mathrm{1.2.3.}}=21$

$\Rightarrow n\frac{n-1}{2}=21$

Solve to get $n=7$