How can you show that each new line drawn on a plane creates another $k+1$ regions?

Asked Jul 03 '17 at 23:22

Active Jul 03 '17 at 23:31

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The questions is

If $n$ straight lines are drawn on a plane such that each line intersects every other line and no three lines have a common point of intersection, then the plane is divided into $$\frac{n(n+1)}{2}+1$$ regions. Prove this by using mathematical induction.

I did the base case, but I do not know how to show that if $P(k)$ is true, then $P(k+1)$ is true. The worked solutions say that the addition of another line creates another $k+1$ regions. How do they get that?

Thanks

edited Jul 03 '17 at 23:31

N. F. Taussig

76,571

asked Jul 03 '17 at 23:22

John Arg

@ThomasGrubb, you're right, thanks, I hadn't seen that – John Arg Jul 03 '17 at 23:31
@N. F. Taussig, how do you make the question like that? – John Arg Jul 03 '17 at 23:32
If you click edit (at the bottom of your question), you can see how I did it. Also, you may find it useful to read this tutorial on how to typeset mathematics on this site. – N. F. Taussig Jul 03 '17 at 23:34
@N.F.Taussig Thanks, I'll look at that – John Arg Jul 03 '17 at 23:37

How can you show that each new line drawn on a plane creates another $k+1$ regions?

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