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I've always been a bit iffy about those "Find the next number in the sequence A, B, C, D, ???" problems.

Given any finite sequence of numbers, there will (probably?) be an infinite number of general forms satisfying that finite sequence.

My question is:

Given an arbitrary sequence of numbers, how would one construct a general form that 'forcefully' satisfies it for the given terms, but then goes haywire beyond the given terms?

So for example, consider the sequence 0, 0, 0, 0, 0, ...

The general population would probably say "Well the next term is obviously zero" but it could very well be 120 using the general term

$t_n=n^5-15n^4+85n^3-225n^2+274n-120$

which is clearly zero for the first five terms.

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