Sequence and Series Calculator
Paste the terms you were given and it works out the rule — arithmetic, geometric or quadratic — or build a sequence from its parameters and watch the partial sums converge.
aₙ = 3n − 1
How it was identified
- Take the differences between consecutive terms3, 3, 3, 3
- The differences are all the same, so this is arithmeticThe common difference is d = 3. The nth term is a + (n − 1)d, which expands to 3n − 1.
Terms and running total
| n | Term | Sum of the first n |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 5 | 7 |
| 3 | 8 | 15 |
| 4 | 11 | 26 |
| 5 | 14 | 40 |
| 6 | 17 | 57 |
| 7 | 20 | 77 |
| 8 | 23 | 100 |
| 9 | 26 | 126 |
| 10 | 29 | 155 |
Paste the terms, not the type
A question that says “find the nth term of 2, 5, 8, 11” is asking you to work out what kind of sequence it is. A calculator whose first control is an arithmetic-or-geometric dropdown has asked you for the answer before it will help, which is backwards.
So the first tab takes the terms. Three tests run in order, each more general than the last:
- Constant first difference → arithmetic. The differences of 2, 5, 8, 11 are 3, 3, 3.
- Constant ratio → geometric. The ratios of 3, 6, 12, 24 are 2, 2, 2.
- Constant second difference → quadratic. The differences of 2, 6, 12, 20, 30 are 4, 6, 8, 10, whose differences are 2, 2, 2.
That third case is a standard exam topic and most calculators skip it entirely. The trick worth remembering: for an² + bn + c, the second difference is always 2a — so half the second difference gives you a immediately, and the first two terms give you b and c.
If none of the three fits, the tool says so rather than forcing the nearest match. Fibonacci, the cubes and the factorials all fail all three tests while following perfectly good rules.
Why exact arithmetic matters here
The identification tests all ask “are these values the same?”, which is a question floating point answers badly. Take the sequence 1/3, 2/3, 1, 4/3. In decimals the differences come out as 0.33333333333333331, 0.33333333333333337 and 0.33333333333333326 — three visibly different numbers, and a tool comparing them has to pick a tolerance and hope.
In exact fractions all three differences are 1/3, the sequence is arithmetic, and there is nothing to decide. The nth term comes out as n/3, and the twentieth term is exactly 20/3.
The formulas, and when each applies
Arithmetic
The nth term is a + (n − 1)d, which expands to dn + (a − d) — and it is worth writing it the second way, because that is the form an exam answer usually wants: a linear expression in n. The sum of the first n terms is (n/2)(2a + (n − 1)d).
Geometric
The nth term is arn−1. The sum of the first n terms is a(1 − rn)/(1 − r), which has a hole in it at r = 1 — the denominator vanishes. That case is handled separately here, because the sequence is then just a repeated n times and the sum is na. A tool that divides anyway returns NaN or Infinity for a question with a perfectly ordinary answer.
Quadratic
The nth term is an² + bn + c, with a equal to half the second difference. There is a closed form for the sum too, but it is rarely asked for, so the partial sums here are accumulated directly.
The sum to infinity
Adding infinitely many things can give a finite answer, and a geometric series is where that is first met. It converges exactly when |r| < 1, and the sum is a/(1 − r).
The two boundary cases fail for genuinely different reasons, and this tool distinguishes them rather than printing “diverges” twice:
- r = 1. Every term is the same, so the partial sums are a, 2a, 3a, … and grow without bound.
- r = −1. The partial sums alternate between a and 0 forever. They stay bounded — they never grow — but they never approach a single value either, and approaching a single value is what convergence means. This is the case that makes people think 1 − 1 + 1 − 1 + … equals 1/2.
When a series converges, the plot draws its limit as a horizontal guide so you can watch the partial sums flatten against it.
Four mistakes this will catch
- Using n rather than n − 1 in the nth term. The first term is a, not a + d. Every formula here is 1-indexed and the table shows n beside each term so you can check.
- Confusing the sequence with the series. The sequence is the list of terms; the series is their sum. A sequence can converge to zero while its series diverges — the harmonic sequence 1, 1/2, 1/3, … does exactly that.
- Assuming shrinking terms mean a convergent series. Necessary, not sufficient. For a geometric series it happens to be enough, which is why the habit forms and then fails elsewhere.
- Reading a pattern from too few terms. 1, 2, 4 is geometric; 1, 2, 4, 7 is quadratic. The tool warns when it has only three terms to go on.
What it does not do
- Recursive definitions such as Fibonacci, where each term depends on the ones before it.
- Sequences given by a formula you type, rather than by their terms or parameters.
- Convergence tests for non-geometric series — ratio, comparison, integral.
- Cubic and higher polynomial sequences.
- More than 60 terms at a time.
Questions
How does it know whether my sequence is arithmetic or geometric?
It takes the differences between consecutive terms and checks whether they are all the same; if they are, the sequence is arithmetic. If not, it takes the ratios and checks those. If neither is constant it takes the differences of the differences, and a constant second difference means a quadratic sequence. Every test is exact, which matters for fractions — 1/3, 2/3, 1 is arithmetic, and a tool working in decimals has to decide whether 0.33333333 and 0.33333334 are the same difference.
Why does it refuse Fibonacci instead of giving me a formula?
Because Fibonacci is none of the three patterns it tests for. Its differences are not constant, its ratios are not constant, and its second differences are not constant either. There is a closed form — Binet’s formula, built on the golden ratio — but it is not something these three tests can find, and inventing a nearest fit would be worse than saying so. The result names Fibonacci, cubes and factorials as sequences that fail all three tests but still follow a rule.
Is three terms enough to identify a sequence?
It is the minimum, and it is thin evidence. Two terms have exactly one difference and one ratio, so they fit both an arithmetic and a geometric rule and identify nothing. Three terms can show a constant difference, but a fourth term could break it — 1, 2, 4 is geometric, and 1, 2, 4, 7 is quadratic. The tool says when it worked from only three terms.
What makes a geometric series converge?
The terms have to shrink fast enough, which for a geometric series means |r| < 1. Then the sum to infinity is a/(1 − r), exactly. The two boundary cases fail for different reasons and the tool distinguishes them: r = 1 makes every term the same so the partial sums grow without bound, while r = −1 makes the partial sums alternate between a and 0 — bounded, but never approaching a single value, which is what convergence actually requires.
Why are the terms drawn as separate dots rather than joined up?
Because a sequence is only defined at whole numbers. There is no third-and-a-half term, and a line between consecutive dots would claim there is. The partial sums do get a joining line, because the running total is a process with an order to it and the line is what makes the trend — flattening towards a limit, or climbing away from one — visible.
Does it keep fractions exact?
Throughout. A sequence with a common difference of 1/3 has a tenth term of exactly 10/3, not 3.3333333333333335, and a geometric series with r = 1/2 sums to exactly 2. Nothing here converts to a floating-point number along the way.