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In a post on Math.SE., a proof sketch was proposed for the proposition below:

The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$.

Anyone familiar with algebraic identities and induction can understand that proof, but it's not the case for a 7th-grade students!

Is there an elementary proof for this proposition?

Thanks.

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Behzad
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  • There is a purely combinatorial argument, which I give here. With a lot of effort and simplification, and lots of examples and a slow build-up I think that might be accessible, or at least vaguely understandable for a talented 13-year-old. – Izaak van Dongen Nov 05 '23 at 16:22

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