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 :
7
Solution :
$$\therefore \,\,{T_n} = {\,^n}{C_3}\,;\,{T_{n + 1}} = {\,^{n + 1}}{C_3}$$
As per question,
$$\eqalign{
& {T_{n + 1}} - {T_n} = 21 \cr
& \Rightarrow \,{\,^{n + 1}}{C_3}{ - ^n}\,{C_3} = 21 \cr
& \frac{{\left( {n + 1} \right)n\left( {n - 1} \right)}}{{3.2.1}} - \frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{3.2.1}} = 21 \cr
& \Rightarrow \,\,n\left( {n - 1} \right)\left( {n + 1 - n + 2} \right) = 126 \cr
& \Rightarrow \,\,n\left( {n - 1} \right) = 42 \cr
& \Rightarrow \,\,n\left( {n - 1} \right) = 7 \times 6 \cr
& \Rightarrow \,\,n = 7. \cr} $$