An Algebra Curiosity

By Dr. Mike Ecker

If you take the product of any three consecutive integers and then add the middle number, the result is always a perfect cube (of the middle number).

For example,
3*4*5 + 4 = 64 = 43
4*5*6 + 5 = 125 = 53

Why does this work?

Use some algebra to prove it:
(n - 1) n (n + 1) = n(n2 - 1) + n = n3 - n + n = n3

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