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Expression Formation

Let Tn denote...

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

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$T h e n u m b e r o f t r i a n g l e s t h a t c a n b e f o r m e d b y u \sin g t h e v e r t i c e s o f a r e g u l a r p o l y g o n i s^{n} C_{3} . T h a t i s T_{n} =^{n} C_{3} N o w, T_{n + 1} - T_{n} = 21 \Rightarrow^{n + 1} C_{3} -^{n} C_{3} = 21 \Rightarrow^{n} C_{2} +^{n} C_{3} -^{n} C_{3} = 21 [∵^{n + 1} C_{r} =^{n} C_{r - 1} +^{n} C_{r}] \Rightarrow \frac{1}{2} n (n - 1) = 21 \Rightarrow n = - 6 o r 7 A s n i s a p o s i t i v e i n t e g e r, n = 7$
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