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I'm stumped on this problem: Let [itex]R_n[/itex] denote the nubmer of regions into which the plane is divided by [itex]n[/itex] lines. Assume that each pair of lines meets in a point, but that no three lines meet in a point. Derive a recurrence relation for the sequence [itex]R_1[/itex], [itex]R_2[/itex], ....

After doing a couple of examples, the recurrence relation seems to be

[tex]R_n = R_{n-1} + n[/tex]. I want to prove that this is the proper recurrence relation. More specifically, I want to show that when I cut a plane consisting of [itex]R_k[/itex] regions with the appropriate line, [itex]k+1[/itex] of those regions will be divided in two so that the number of regions is [itex]R_{k+1} = R_k + k + 1[/itex]. I can't think of any geometrical argument for this. Any tips?

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# Number of Regions in a Plane

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