Question

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon on 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 $7$

Number of triangles that can be formed by using the vertices of a regular polygon$={n_c}_3$

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

${{n+1}_c}_3-{n_c}_3=21$

$\Rightarrow {n_c}_2+{n_c}_3-{n_c}_3=21$ since ${{n+1}_c}_r={n_c}_{r-1}+{n_c}_r$

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

$\Rightarrow n^2-n=42$

$\Rightarrow n^2-n-42=0$

$\Rightarrow n^2-7n+6n-42=0$

$\Rightarrow n(n-7)+6(n-7)=0$

$\Rightarrow n+6=0,n-7=0$

$\therefore n=-6,7$

As $n$ is a positive integer, we have $n=7$