A worked example: generating with Sequence, detecting with Number Sequence
Set the Sequence Calculator to arithmetic with a first term of 7 and a common difference of 4, and generating terms 1 through 6 gives 7, 11, 15, 19, 23, 27. Take just the first four of those — 7, 11, 15, 19 — and type them into the Number Sequence Calculator instead. It detects patternType "arithmetic", a common value of 4, and predicts the next term as 23. That's exactly what the Sequence Calculator itself returns when asked directly for n = 5 with the same first term and common difference: an nth term of 23 (and a sum of the first five terms of 75). The two calculators start from opposite ends — one guesses the rule from data, the other applies a rule you already specified — and land on the same number.
The formula: two different directions through the same math
Number Sequence takes a free-text list of observed numbers as its only input. It checks for a constant difference between consecutive terms first, then a constant ratio, and outputs a pattern label plus a single predicted next term — it never asks you what kind of sequence you expect. Sequence takes the opposite input shape: an explicit sequence type (arithmetic or geometric), a first term, a common difference or ratio, and a term number n — then evaluates the closed-form nth-term and sum formulas directly. It performs no detection at all; you have to already know the type. The same round trip works for geometric sequences: Sequence with a first term of 3 and a common ratio of 2 generates 3, 6, 12, 24, 48, 96 for n = 1 through 6. Feeding the first four terms, "3, 6, 12, 24", into Number Sequence detects patternType "geometric" with a common value of 2 and a predicted next term of 48 — matching Sequence's own nth term at n = 5 (48, with a sum of the first five terms of 93).
Where they diverge: zero as a first term
Sequence enforces a hard validation rule: a geometric sequence cannot start at 0, and asking for one raises the error "The first term cannot be zero for a geometric sequence." Arithmetic sequences face no such restriction — a first term of 0 with a common difference of 3 computes normally, returning an nth term of 12 and a sum of the first five terms of 30 at n = 5. Number Sequence never raises an error over a zero at all. Instead, its geometric ratio check is silently skipped whenever any entered number is zero, since a ratio computed through zero is either undefined or degenerate. Scanning 5, 0, -5, -10 simply falls back to the arithmetic check, which matches, reporting a common value of -5 and a predicted next term of -15. And unlike Sequence, Number Sequence can genuinely fail to find any pattern — feed it 1, 2, 4, 7, 11 (differences of 1, 2, 3, 4; ratios that aren't constant either) and it reports patternType "unknown" with an explanatory message. Sequence has no equivalent failure mode, because it never attempts to guess a pattern in the first place — it only ever computes the one you told it to.
Why picking the right direction matters
If you're staring at a handful of observed numbers and trying to work out the underlying rule, Number Sequence is the right tool — it does the detective work of checking constant differences and ratios for you. If you already know the type of sequence and need a specific far-off term or a running total without listing every value in between, Sequence is the right tool — its closed-form formulas get you there in one step, for any term number up to 1,000. Reaching for the wrong one means either typing out a rule you haven't confirmed, or hand-checking a pattern the other calculator would have verified instantly.