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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: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond Re: Number of Squares proof February 3rd, 2013, 04:06 AM   #3
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Re: Number of Squares proof

Quote:
 Originally Posted by Jakarta (n(n+1)(2n+1))/6
That's the standard sum of squares formula. February 3rd, 2013, 08:30 AM   #4
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Re: Number of squares proof

Quote:
 Originally Posted by Jakarta 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
The inductive step is as follows.

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 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. Tags number, proof, squares Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jjvivde Real Analysis 1 January 8th, 2014 11:05 PM MrBibbles Number Theory 4 July 25th, 2013 06:46 AM thenight Complex Analysis 1 March 27th, 2013 02:50 AM icemanfan Number Theory 2 November 15th, 2012 08:41 AM momesana Algebra 1 March 13th, 2010 03:16 PM

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