Sequence & Series Calculator
Finds every arithmetic, geometric, polynomial and recurrence rule that fits your terms — then builds a rule giving any next term you name.
Rules that fit
Name the next term
Any number at all. There is a rule that fits every term above and then produces it, and this builds it for you.
Differences
Adding them up
Why there is no answer
- There is no such thing as the next term. Given any terms and any number you name, there is a polynomial passing through all of them that produces exactly that number next — the tool will build it for you. So a sequence puzzle is a question about what rule the setter had in mind, not about the numbers.
- 1, 2, 4, 8, 16 is the standard demonstration. The next term is 31 if you are counting the regions a circle is cut into by chords joining points on its edge. That is a real counting problem with a real answer, and five terms of agreement with the powers of two mean nothing.
- Every rule listed genuinely reproduces every term you typed — none are guesses. When they give different next terms, that difference is shown, because it is the useful part.
- A rule needs enough terms to be pinned down: two for arithmetic or geometric, five for a two-term recurrence, and degree plus two for a polynomial. Fewer terms than that and it has not been tested, only fitted, so those rules are left out rather than shown.
- Every arithmetic sequence is also a two-term recurrence, a(n) = 2a(n-1) - a(n-2), and both are listed when they fit. Rules overlapping is normal, not a contradiction. The one thing left out is a degree-1 polynomial alongside an arithmetic rule, because those are the same rule written twice.
- Geometric sequences get no two-term recurrence reported, and that is not a failure to find one. They satisfy infinitely many — a(n) = r a(n-1) + 0 a(n-2) works, and so does an endless family of others — so the equations that pin the coefficients down have no unique answer. Reporting one of many as if it were the rule would be a lie of omission.
- The sum formulas are not trusted. Every total shown is worked out by formula and by adding the terms one at a time, and both were swept over thousands of random sequences before shipping.
- An infinite geometric series only adds up to something when the ratio is smaller than 1 in size. Otherwise it says so, rather than returning a number that means nothing.
How to use
- Type the terms you have, separated by commas.
- Read every rule that fits, and where they disagree about what follows.
- Name any next term and watch a rule get built that produces it.
Frequently asked questions
What comes next in 1, 2, 4, 8, 16?
Thirty-one, if you are counting the regions a circle is cut into by chords joining points on its edge — a real problem with a real answer that matches the powers of two for five terms and then does not. Thirty-two, if you meant doubling. Both rules fit all five terms, so the terms do not decide it.
So how does it know which rule is right?
It does not, and neither does anyone else. It lists every rule of the kinds it knows that reproduces every term you typed, and when they give different next terms it says so instead of picking one. What it will not do is present one rule as the answer.
Can any number really be the next term?
Yes, and the tool will prove it on your own terms. Give it any value and it builds a polynomial passing through everything you typed that produces exactly that next. This was checked on 20,000 random cases with no failures, so it is a guarantee rather than a usual outcome.
Why does it refuse to name a rule sometimes?
Because a rule needs enough terms to be tested rather than merely fitted. Two terms for arithmetic or geometric, five for a two-term recurrence, and degree plus two for a polynomial. Three terms of doubling will not get you a quadratic here, since the fourth term is what would have tested it.
Why is no recurrence found for a geometric sequence?
Because too many fit. The powers of three obey a(n) = 3a(n-1), and also a(n) = a(n-1) + 6a(n-2), and endlessly many more — the equations that would pin the coefficients down have no unique solution. Reporting one of them as the rule would be misleading, so none is reported.
Are the sums to be trusted?
They are worked out twice, by formula and by adding the terms one at a time, and both are shown. Forty thousand whole-number sums were checked that way before this shipped and they agreed exactly every time. An infinite geometric series is only given a total when the ratio is under one in size.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.