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April 28th, 2015, 02:07 AM  #1 
Newbie Joined: Apr 2015 From: uk Posts: 1 Thanks: 0  cubed number sequence
the sequence of cubed numbers increases by a sequence of numbers that differ by the six times table ie 0,1,8,27,64 1,7,19,37 (the difference) 6,12,18,24 (the second row of difference) my question is why is that?! 
April 28th, 2015, 03:04 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,474 Thanks: 2039 
As the original numbers are cubes, the first differences are given by a quadratic expression and the second differences by a linear expression. It happens to be the case that this linear expression has no constant term, so the second differences form a times table. Had you started with cube plus square, you would have got 0, 2, 12, 36, 80, . . . 2, 10, 24, 44, . . . (first differences) 8, 14, 20, . . . (second differences) so the second differences no longer form a times table. Note that if you start with nth powers, the nth differences all equal n factorial. 

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