Let $T_{n}$ denotes 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

Question

Dr. Abubacker · Accepted Answer

(b)
According to given condition, $T_{n} = \ ^{n}C_{3}$
and $T_{n + 1} - T_{n} = 21$
$\Rightarrow \ ^{n + 1}C_{3} - \ ^{n}C_{3} = 21$
$\Rightarrow \ \frac{1}{6}(n + 1)(n)(n - 1) - \frac{1}{6}n(n - 1)(n - 2) = 21$
$\Rightarrow \ \frac{n(n - 1)}{6}\left\lbrack (n + 1) - (n - 2) \right\rbrack = 21$
$\Rightarrow \ \frac{n(n - 1) \bullet 3}{6} = 21$
$\Rightarrow \ n(n - 1) = 42$
$\Rightarrow \ n = 7$

Q. Let $T_{n}$ denotes 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

Explanation

Reviewed by Dr. Abubacker

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