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: 7,852 Thanks: 1077 Math Focus: Elementary mathematics and beyond  Re: induction
Induction hypothesis: The number of regions is Show it holds for n = 1: Use the induction hypothesis to find the difference between consecutive numbers of regions: Induction: 
September 28th, 2011, 07:55 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,505 Thanks: 1741 
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 n1 lines.

September 28th, 2011, 08:43 PM  #4  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,852 Thanks: 1077 Math Focus: Elementary mathematics and beyond  Re: induction Quote:
or  
September 28th, 2011, 08:55 PM  #5  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,852 Thanks: 1077 Math Focus: Elementary mathematics and beyond  Re: induction Quote:
 

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