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## Regions Made by Intersecting Lines

Logic, Pattern, and Formula for maximum number of regions that can be made by intersection of n lines in a plane.

**Questions:**

What is the maximum number of regions created by intersection of 5 lines in a plane?

How many regions (maximum) will 6 intersecting lines create in a plane ?

**One line** creates 2 regions in a plane (whichever way we draw it)

Total regions =2

**2 ^{nd} line** creates 2 more regions in the plane when it intersects with the 1st line.

Total regions = 2+2=4

For creating maximum number of regions , point of intersection of each pair of lines should be different.

**3 ^{rd} line** can create maximum 3 more regions in the plane when it intersects with the

existing 2 lines.

Total regions = 3+2+2=7

**4 ^{th} line** can create maximum 4 more regions in the plane when it intersects with the

existing 3 lines.

Total regions = 4+3+2+2=11

Similarly,**5th line** will create total = 5+4+3+2+2 = 16 regions**6th line** will create total = 6+5+4+3+2+2 =22 regions

Hence, every nth line can create maximum n number of extra regions in a plane when it intersects with the existing (n-1) lines.

Hence n lines will create maximum n+(n-1)+(n-2)……..3+2+2 regions

= n(n+1) + 1 Regions

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