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}}=28, $ then $ n $ equals

• A. 4
• B. 5
• C. 6
• D. 7
• E. 8

Answer 1

Answer: (E)

Given, $ {{T}_{n+1}}-{{T}_{n}}=28 $
$ \Rightarrow $ $ ^{n+1}{{C}_{3}}{{-}^{n}}{{C}_{3}}=28 $
$ \Rightarrow $ $ \frac{(n+1)!}{3!(n-2)!}-\frac{n!}{(n-3)!3!}=28 $
$ \Rightarrow $ $ \frac{1}{6}[(n+1)(n)(n-1)-n(n-1)(n-2)]=28 $
$ \Rightarrow $ $ n[{{n}^{2}}-1-({{n}^{2}}-3n+2)]=168 $
$ \Rightarrow $ $ {{n}^{2}}-n-56=0 $
$ \Rightarrow $ $ n=8 $ $ (n\ne -7) $