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What is the maximum number of regions into which n chords can divide a circle?

I have gotten all my data, and I am having trouble with writing the equation.

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I notice that the first difference is not constant, but the second difference is. The second difference is +1. I know that this is a quadratic equation.

I believe the equation is r=(n(n+1))/2, if r=maximum number of regions. However, using that equation I am off by 1. Where am I going wrong? Can someone walk me through how to write the equation correctly?

Dana
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  • Why not just add $1$? I'm not sure if this equation is right, but $\frac{n(n+1)}{2}+1$ seems to fit that data set. If you want to prove it, you might do some kind of induction argument. – user153582 Oct 30 '15 at 03:26
  • So I figured out an equation that does work, r=(n(n-1)/(2+1), but why does it work? – Dana Oct 30 '15 at 03:49
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    The formula should be $ \frac {n^2 +n +2}{2}$ as pointed out above. See: http://mathworld.wolfram.com/CircleDivisionbyLines.html for more information. – Nicholas Oct 30 '15 at 03:57
  • I'm still a bit confused as to how we got from f(2)=2+f(1) to f(n)=n+f(n-1). – Dana Oct 30 '15 at 04:15
  • As a general rule in life, whenever dealing with integer sequences, always check OEIS. It contains a wealth of information and references. – Lucian Oct 30 '15 at 06:48

1 Answers1

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when you add the kth line there already exist k-1 lines so there are at most k-1 new intersections you can create. these k-1 new intersections create k portions on the kth line hence k new regions in the circle so $f(k)=f(k-1)+k$. summing both sides from k=1 to n yields $f(n)=f(0)+n(n+1)/2$. since $f(0)=1$ we have $f(n)=(n^2+n+2)/2$

nonuser
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peter leung
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