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 September 28th, 2011, 04:56 AM #1 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 induction How to logically explain this induction case? If n straight lines are drawn such that each line intersects every other line and no three lines have a common point of intersection, then the plane is divided into (n(n+1))/2 +1 regions. I would appreciate a clear explanation, thank you.
 September 28th, 2011, 06:52 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,814 Thanks: 1046 Math Focus: Elementary mathematics and beyond Re: induction Induction hypothesis: The number of regions is $\frac{n(n\,+\,1)}{2}\,+\,1$ Show it holds for n = 1: $\frac{1(2)}{2}\,+\,1\,=\,2$ Use the induction hypothesis to find the difference between consecutive numbers of regions: $\frac{n(n\,+\,1)}{2}\,+\,1\,-\,$$\frac{n(n\,-\,1)}{2}\,+\,1$$$ $\frac{n(n\,+\,1)}{2}\,+\,1\,-\,\frac{n(n\,-\,1)}{2}\,-\,1\,=\,\frac{n^2\,+\,n\,-\,n^2\,+\,n}{2}\,=\,n$ Induction: \begin{align*}\frac{n(n\,+\,1)}{2}\,+\,1\,+\,(n\,+ \,1)\,&=\,\frac{n^2\,+\,n\,+\,2\,+\,2n\,+\,2}{2} \\ &=\,\frac{n^2\,+\,3n\,+\,2}{2}\,+\,1 \\ &=\,\frac{(n\,+\,1)(n\,+\,2)}{2}\,+\,1\end{alig n*}
 September 28th, 2011, 07:55 PM #3 Global Moderator   Joined: Dec 2006 Posts: 19,045 Thanks: 1618 That's effectively doing the induction step twice. Instead, claim that it can be "seen" that the nth line adds n regions to the number of regions generated by n-1 lines.
September 28th, 2011, 08:43 PM   #4
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Re: induction

Quote:
 Originally Posted by skipjack Instead, claim that it can be "seen" that the nth line adds n regions to the number of regions generated by n-1 lines.
\begin{align*}\frac{n(n\,-\,1)}{2}\,+\,1\,+\,n\,\,&=\,\frac{1(1\,-\,1)}{2}\,+\,2\,\leftarrow\,n\,=\,1 \\
&=\,2\end{align*}

$\frac{n(n\,-\,1)}{2}\,+\,1\,+\,n\,=\,\frac{n^2\,-\,n\,+\,2\,+\,2n}{2}\,=\,\frac{n(n\,+\,1)}{2}\,+\, 1$

or

$\frac{n(n\,+\,1)}{2}\,+\,1\,+\,(n\,+\,1)\,=\,\frac {n^2\,+\,n\,+\,2\,+\,2n\,+\,2}{2}\,=\,\frac{n^2\,+ \,3n\,+\,2\,+\,2}{2}\,=\,\frac{(n\,+\,1)(n\,+\,2)} {2}\,+\,1$

September 28th, 2011, 08:55 PM   #5
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Re: induction

Quote:
 Originally Posted by skipjack That's effectively doing the induction step twice.

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