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This question already has an answer here:

Suppose we we have n straight lines on the plane such that no two of them are parallel and no three of them go through the same point. Prove that the number of different regions that are created by these lines (regions that are bounded between line segments and/or those that are unbounded) is exactly $\frac{n(n + 1)}{2} + 1$ for all n ≥ 1.

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marked as duplicate by Brian M. Scott discrete-mathematics Mar 12 '15 at 4:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ You may also find this question and this one (and their answers) useful. $\endgroup$ – Brian M. Scott Mar 12 '15 at 4:28
  • $\begingroup$ in base case if n holds for 1 and 2, then we can conclude that p(n) holds, $\endgroup$ – Tiina Mar 12 '15 at 4:28