Maximum number of regions n lines can dissect a plane
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
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
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