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Sum of Cubes (Posted on 2004-05-25) Difficulty: 3 of 5
Prove that the sum of consecutive perfect cubes (starting with 1) is always a perfect square.

For example:
1=1
1+8=9
1+8+27=36

See The Solution Submitted by Gamer    
Rating: 3.4000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
What Gamer perhaps had in mind. Comment 11 of 11 |

Not just any old perfect square, but the square of the nth triangular number, whereupon the proof becomes easy:

 

I    SIGMA(x=1 to x) x^3 is 1/4*x^2*(x+1)^2

II   (n(n+1)/2)^2=1/4*n^2*(n+1)^2

 

And the two expressions are clearly equivalent.

 

QED

Edited on October 10, 2011, 2:40 am
  Posted by broll on 2011-10-10 02:38:30

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