n-lines-are-drawn-inside-a-circle-in-such-a-way-that-the-circle-has-been-divided-in-maximum-number-of-parts-Determine-this-maximum-number-

Question Number 2642 by Rasheed Soomro last updated on 24/Nov/15

n 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 s u c h a w a y t h a t

t h e c i r c l e h a s b e e n d i v i d e d i n 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 m a x i m u m n u m b e r .

Commented by RasheedAhmad last updated on 24/Nov/15

∙ O n e l i n e c a n d i v i d e t h e c i r c l e

a t m o s t 2 p a r t s .

∙ T w o l i n e s c a n d i v i d e t h e c i r c l e

a t m o s t 4 p a r t s .

∙ T h r e e l i n e s c a n d i v i d e t h e c i r c l e

i n a t m o s t 7 p a r t s .

\dots .

\dots

∙ n l i n e s c a n d i v i d e t h e c i r c l e i n

a t m o s t (s a y) m p a r t s .

\frac{W h a t i s m ?}{}

Answered by prakash jain last updated on 24/Nov/15

n lines will divide a circle is maxium number

of parts when

∙ 3 or more lines are not concurrent

∙ All intersection points are inside the circle

Let a_{i} be the number of parts circle

is divided into after drawing i^{t h} line .

a_{0} = 1 (n o l i n e s a r e d r a w n)

a_{1} = a_{0} + 1

a_{2} = a_{1} + 2

a_{3} = a_{2} + 3

a_{4} = a_{3} + 4

\dots

a_{n} = a_{n - 1} + n

m = a_{n} = 1 + \frac{n (n + 1)}{2}

Commented by Rasheed Soomro last updated on 25/Nov/15

E x c e l l e n t S i r!