Generate arithmetic, geometric, Fibonacci and custom-rule sequences with the nth-term formula, partial sums, and a live chart.
Pick a sequence type and enter parameters — terms update live.
Generated sequence
Terms (n)
10
count generated
nth term
28
last value aₙ
Sum Sₙ
145
a₁ + a₂ + … + aₙ
Mean
14.5
Sₙ / n
| Type | nth-term formula | Sum Sₙ |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n − 1)·d | Sₙ = n(a₁ + aₙ) / 2 |
| Geometric | aₙ = a₁ · r^(n − 1) | Sₙ = a₁ · (1 − rⁿ) / (1 − r), r ≠ 1 |
| Fibonacci | Fₙ = Fₙ₋₁ + Fₙ₋₂ | ΣFₖ = Fₙ₊₂ − 1 (when F₁=1) |
| Quadratic | aₙ = An² + Bn + C | Σ closed form via Faulhaber |
| Triangular | Tₙ = n(n + 1)/2 | 1, 3, 6, 10, 15, 21… |
| Square | Sₙ = n² | 1, 4, 9, 16, 25, 36… |
The Method
A sequence is an ordered list of numbers built by a rule. The two simplest and most common rules are add a constant (arithmetic) and multiply by a constant (geometric). Both have neat closed-form expressions for the nth term and the partial sum. The Fibonacci sequence — each term is the sum of the two before — generates one of the most studied sequences in mathematics, with surprising appearances in nature, art and finance.
Working for current sequence
About This Tool
A number sequence calculator generates a list of numbers built by a rule, then computes the nth-term formula, the partial sum, the mean, and a chart of the terms. Five rules are built in: arithmetic (add a constant), geometric (multiply by a constant), Fibonacci (sum the two preceding terms), quadratic (an² + bn + c), and custom list (paste any data and inspect it as a sequence).
Sequences are everywhere in mathematics. Arithmetic and geometric sequences underpin compound interest, population growth, annuity payments, and discrete-time signal processing. The Fibonacci sequence appears in phyllotaxis (the arrangement of leaves on a stem), in Fibonacci retracements used by traders, and in algorithmic complexity. Quadratic sequences describe polygonal numbers (triangular, square, pentagonal), and many physical processes with quadratic time dependence.
This calculator uses standard closed-form formulas. Arithmetic: aₙ = a₁ + (n − 1)d, Sₙ = n(a₁ + aₙ) / 2. Geometric: aₙ = a₁·r^(n − 1), Sₙ = a₁·(1 − rⁿ) / (1 − r) when r ≠ 1. Fibonacci uses the recurrence directly. Each value is evaluated with double-precision arithmetic — accurate to about 15 significant digits.
Use this free number sequence calculator for homework, sequence-and-series revision, programming interview prep, or any time you need to inspect or generate a list of numbers built from a rule. Calculation runs locally — no sign-up, no tracking.
Five Sequence Types
Arithmetic, geometric, Fibonacci, quadratic and free-form custom lists.
Closed-Form nth Term
Displays the formula so you can compute any future term by hand.
Partial Sum Sₙ
Returns the sum of the generated terms in closed form when possible.
Term-by-Term Plot
SVG chart shows the shape of the sequence — linear, exponential, or polynomial.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Copy as Comma List
One-click copy of the generated values for use in Python, Excel, etc.
From rule to chart in seconds.
Tap one of the chips at the top of the card — Arithmetic, Geometric, Fibonacci, Quadratic, or Custom list.
Fill in the rule's parameters — first term, common difference/ratio, term count, or quadratic coefficients.
The generated comma-separated list appears in the summary card. Stat tiles show n, aₙ, Sₙ, and mean.
The SVG plots aₙ vs n so you can spot linear vs exponential vs polynomial growth at a glance.
The formula card shows the nth-term formula with substituted values — compute any future term without re-running.
Use the Copy button to grab the comma-separated values for use in Python, Excel, or a notebook.
Everything you need to know about number sequences and series.
An arithmetic sequence adds a fixed common difference d to get from one term to the next. The nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n(a₁ + aₙ) / 2. Example: 3, 7, 11, 15, 19… with a₁ = 3, d = 4.
A geometric sequence multiplies by a fixed common ratio r each step. The nth term is aₙ = a₁·r^(n − 1) and the sum is Sₙ = a₁·(1 − rⁿ) / (1 − r) for r ≠ 1. Example: 2, 6, 18, 54… with a₁ = 2, r = 3.
The Fibonacci sequence starts 0, 1 and every subsequent term is the sum of the two before: Fₙ = Fₙ₋₁ + Fₙ₋₂. So 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… The ratio between consecutive terms converges to the golden ratio φ ≈ 1.6180339887.
φ = (1 + √5) / 2 ≈ 1.6180339887. It satisfies φ² = φ + 1 and is the limit of ratios Fₙ₊₁/Fₙ in the Fibonacci sequence. Closed-form for Fₙ (Binet's formula): Fₙ = (φⁿ − ψⁿ) / √5, where ψ = 1 − φ ≈ −0.618.
A quadratic sequence has nth term of the form aₙ = An² + Bn + C. Differences between consecutive terms grow linearly, and second differences are constant (= 2A). Examples include the triangular numbers 1, 3, 6, 10, 15… and square numbers 1, 4, 9, 16, 25…
A sequence is an ordered list of numbers (a₁, a₂, a₃, …). A series is the sum of a sequence's terms (a₁ + a₂ + a₃ + …). So "the Fibonacci sequence" lists the numbers; "the Fibonacci series" sums them.
An infinite geometric series converges if and only if |r| < 1. The infinite sum is then S∞ = a₁ / (1 − r). If |r| ≥ 1 the terms don't shrink to zero and the sum diverges (or oscillates).
Compound interest is a geometric sequence: each year multiplies by (1 + r). Loan amortisation uses geometric sums. Population growth often follows geometric or logistic models. Computer science uses sequences to analyse algorithms — Fibonacci shows up in tree algorithms; arithmetic sums calculate loop costs. Even music — frequencies of equal-temperament notes form a geometric sequence with ratio 2^(1/12).