Six free-hand lines are drawn inside a circle in order to divide it into maximum number of parts.Determine this number. Note that a free-hand line has following three properties: i) It doesn′t cut itself. ii)It joins two points of circle. iii) It can cut another line at most at three points. Whatif the number of lines is n?


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Question Number 4222 by Rasheed Soomro last updated on 02/Jan/16

$S i x f r e e - h a n d l i n e s a r e d r a w n i n s i d e$ $a c i r c l e i n o r d e r t o d i v i d e i t i n t o m a x i m u m$ $n u m b e r o f p a r t s . D e t e r m i n e t h i s n u m b e r .$ $N o t e t h a t a f r e e - h a n d l i n e h a s f o l l o w i n g$ $t h r e e p r o p e r t i e s :$ $i) I t {d o e s n}^{'} t c u t i t s e l f .$ $i i) I t j o i n s t w o p o i n t s o f c i r c l e .$ $i i i) I t c a n c u t a n o t h e r l i n e a t m o s t a t t h r e e$ $p o i n t s .$ $W h a t i f t h e n u m b e r o f l i n e s i s n ?$
Commented by prakash jain last updated on 03/Jan/16

$a_{0} = 1$ $a_{n} = a_{n - 1} + m (n - 1) + 1$ $a_{n} = 1 + \frac{m (n - 1) n}{2} + n$
Commented by prakash jain last updated on 03/Jan/16

$1 s t line - 2 parts$ $2 nd line - 6 parts$ $n^{t h} line - 3 (n - 1) + 1 parts added$ $a_{0} = 1$ $a_{n} = a_{n - 1} + 3 n - 2$ $a_{n} = 1 + 3 (1) - 2 + 3 (2) - 2 + 3 (3) - 2 + . . . + 3 (n) - 2$ $a_{n} = 1 + \frac{3 n^{2} + 3 n}{2} - 2 n$ $a_{n} = \frac{2 + 3 n^{2} + 3 n - 4 n}{2} = \frac{3 n^{2} - n + 2}{2}$
Commented by Rasheed Soomro last updated on 03/Jan/16

$TH α n k^{S}!$ $W h a t i f t h e f r e e - h a n d l i n e c a n c u t$ $o t h e r l i n e a t m o s t a t m p l a c e s .$
Commented by Rasheed Soomro last updated on 03/Jan/16

$T h α n X again!$
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