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# Maximum number of regions n lines can dissect a plane

Posted on 2008-06-19

I want to find the maximum number of regions that n lines can dissect a plane into.

Let F(n) represent this number.

Through trial and error, I have found that:

F(0)=1

F(1)=2

F(2)=4

F(3)=7

From which I have found a pattern:

F(n) = n + F(n-1) => F(n) = n(n+1)/2 + 1 .

This equation makes some sense, but (assuming that it's accurate for all n), how would I PROVE it to be true?

As all I've done so far, is throw some extrapolation on top of some trial-and-error. Hardly rigorous.

Is such a proof something which I'd be capable of 'getting'? Or, does it depend on some very high-level mathematics?

Thanks