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February 2nd, 2013, 08:00 PM  #1 
Member Joined: May 2012 Posts: 86 Thanks: 0  Number of Squares proof
A square with side length n is divided into n^2 equal squares. Prove by induction that the total number of squares (of any side length) in this array is equal to (n(n+1)(2n+1))/6 Help is appreciated 
February 2nd, 2013, 09:32 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,932 Thanks: 1127 Math Focus: Elementary mathematics and beyond  Re: Number of Squares proof 
February 3rd, 2013, 04:06 AM  #3  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,328 Thanks: 1024  Re: Number of Squares proof Quote:
 
February 3rd, 2013, 08:30 AM  #4  
Member Joined: Jan 2013 Posts: 93 Thanks: 0  Re: Number of squares proof Quote:
Consider an grid of squares obtained by adding to an grid an extra column on the right and an extra row at the top. This creates: additional squares additional squares additional square Total number of additional squares is The number of squares of all sizes in the original grid is by the inductive hypothesis. Hence total number of squares of all sizes in the grid is which greg1313 has shown is equal to . Verify that the formula is true for and you’re done.  

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