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Sum of n Terms

Let T n denot...

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

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Solution

From n non collinear points

Number of triangles formed = $^{n} C_{3} = T_{n}$

So , $^{n + 1} C_{3} -^{n} C_{3} = 21 \Rightarrow (^{n} C_{3} +^{n} C_{2}) -^{n} C_{3} = 21$

$∴$ $^{n} C_{2}$ = 21 or n(n-1) = 42

Roots are 7,-6 . As n can't take negative value.So, n = 7

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