Since there are infinitely many formulas that will fit any finite series, many people object that such problems have no good answer. But isn't this a special case of the general observation that theory is underdetermined by experience (in other words, that there are a lot of world views that are consistent with all the facts that we know)? And if so, doesn't this objection really apply to all puzzles? Isn't it just more obvious in the case of series puzzles?
As a long-time observer of rec.puzzles nit-picking, I have never seen a puzzle answer that could not be challenged. The list of assumptions made in solving any puzzle is neverending. Luckily, most of us share all or nearly all of these assumptions, so that we can agree on an answer when we see it.