Sequence Analyzer & Generating Function

Once any single term breaks the exact arithmetic, geometric, or quadratic pattern check — which a small nudge to a₀ does immediately, since the differences stop matching — the calculator falls back to a formula that only reads a₃ and a₄: twice a₄ minus a₃. a₀, a₁, and a₂ simply don't appear anywhere in that fallback calculation, so changing them has no effect on the prediction once the pattern is broken.

It falls back to a simple linear extrapolation using only the two most recent terms: predicted next term equals twice a₄ minus a₃. This assumes the sequence keeps changing by the same amount it changed between a₃ and a₄, which is a reasonable guess for a roughly linear trend but can be far off for a sequence that's actually curving or oscillating.

Very strict — arithmetic requires all four consecutive differences to be exactly equal, and geometric requires all four consecutive ratios to match to within a tolerance of one billionth. There's no rounding or approximate-match allowance beyond that tiny tolerance, so a sequence that's arithmetic in spirit but off by even a small amount at one step gets routed to a different prediction method entirely.

The fallback prediction formula is twice a₄ minus a₃, so raising a₄ directly increases the result while raising a₃ directly decreases it, since a₃ is being subtracted rather than added. That's the same reasoning behind a straight-line extrapolation: a bigger last term projects the trend further forward, while a bigger second-to-last term implies less recent growth to extend.