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TRManderson comments on Beautiful Math - Less Wrong
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This is obvious after you learn calculus. The "nth difference" corresponds to nth derivative (a sequence just looks at integer points of a real-valued function), so clearly a polynomial of degree n has constant nth derivative. It would be even more accurate to say that an nth antiderivative of a constant is precisely a degree n polynomial.
Neither finite differences nor calculus are new to me, but I didn't pick up the correlation between the two until now, and it really is obvious.
This is why I love mathematics - there's always a trick hidden up the sleeve!