Regions
You have 100 co-planar circles.
What is the maximum number of regions into which the plane can be divided due to their intersections?
Three circles can create up to 7 regions.
Try my set " Let's play with polygons & circles. "
The answer is 9902.
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1 solution
Let f ( n ) represent the maximum number of regions in which a plane is divided by n circles
Then,
f ( n ) = f ( n − 1 ) + 2 ( n − 1 )
f ( n ) − f ( n − 1 ) = 2 ( n − 1 )
on putting n = 2,3,4,...........n
f ( n ) − f ( 1 ) = n ( n − 1 )
We know f ( 1 ) = 2
f ( n ) = n 2 − n + 2
Putting n = 100 , we get 1 0 0 2 − 1 0 0 + 2 = 9 9 0 2