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September 28th, 2011, 05:56 AM   #1
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induction

How to logically explain this induction case?
If n straight lines are drawn such that each line intersects every other line and no three lines have a common point of intersection, then the plane is divided into (n(n+1))/2 +1 regions.

I would appreciate a clear explanation, thank you.
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September 28th, 2011, 07:52 PM   #2
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Re: induction

Induction hypothesis:

The number of regions is

Show it holds for n = 1:



Use the induction hypothesis to find the difference between consecutive numbers of regions:





Induction:

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September 28th, 2011, 08:55 PM   #3
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That's effectively doing the induction step twice. Instead, claim that it can be "seen" that the nth line adds n regions to the number of regions generated by n-1 lines.
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September 28th, 2011, 09:43 PM   #4
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Re: induction

Quote:
Originally Posted by skipjack
Instead, claim that it can be "seen" that the nth line adds n regions to the number of regions generated by n-1 lines.




or

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September 28th, 2011, 09:55 PM   #5
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Re: induction

Quote:
Originally Posted by skipjack
That's effectively doing the induction step twice.
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