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

Question

Dr. Abubacker · Accepted Answer

(b)
$T_{n} = {\ ^{n}C}_{3}$ and $T_{n + 1} - T_{n} = 21$
$\Rightarrow \ \ \ \ \ {\ ^{n + 1}C}_{3} - {\ ^{n}C}_{3} = 21$
$\Rightarrow \ \ \ \ \ \ \ \ {\ ^{n}C}_{2} + {\ ^{n}C}_{3}–{\ ^{n}C}_{3} = 21\$
$\Rightarrow \ \ \ \ \ \ \ \ {\ ^{n}C}_{2} = 21$
$\Rightarrow \ \ \ \ \ \ \ \frac{n(n - 1)}{2} = 21$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ n^{2} - n - 42 = 0$
$\Rightarrow \ \ \ \ \ \ \ \ \ (n - 7)(n + 6) = 0$
$	herefore\ \ \ \ \ \ \ \ \ n = 7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack\because\  
eq - 6brack\ \ \ \ \$

Q. 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

Explanation

Reviewed by Dr. Abubacker

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