Find the nth term, the rule, and the sum of any arithmetic or geometric sequence — or paste the terms you already have and let the calculator detect the pattern for you.
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SEQUENCE ENGINE
TRY
Results update live as you type. Everything is computed locally in your browser — nothing you enter ever leaves this page.
HOW TO USE
From three numbers to the whole sequence.
No calculate button, no waiting — the tool solves as you type.
01
Choose a mode
Arithmetic if the sequence adds the same amount each step, geometric if it multiplies — or find the rule to detect the pattern from terms you already have.
02
Enter your values
First term a₁, the common difference d (or ratio r), and how many terms n you care about. Fractions, decimals and negative numbers are all welcome.
03
Read the results
The rule, the nth term with the working shown, the term list, the sum of the first n terms — and a chart of how the sequence grows.
04
Use the extras
Type any value into the is this in the sequence? box to find which term it is — then hit copy to take the whole solution with you.
CHEAT SHEET
The formulas, at a glance.
Arithmetic sequences
constant difference d — e.g. 2, 5, 8, 11…
nth termaₙ = a₁ + (n − 1)d
sum of n termsSₙ = n/2 · (2a₁ + (n − 1)d)
sum, other formSₙ = n/2 · (a₁ + aₙ)
mean of terms(a₁ + aₙ) / 2
Geometric sequences
constant ratio r — e.g. 3, 6, 12, 24…
nth termaₙ = a₁ · rⁿ⁻¹
sum of n termsSₙ = a₁(1 − rⁿ)/(1 − r)
infinite sumS∞ = a₁/(1 − r)
convergenceinfinite sum exists only if |r| < 1
FAQ
Sequence calculator — questions.
What is an arithmetic sequence?
An arithmetic sequence adds the same number — the common difference d — at every step: 2, 5, 8, 11, 14… has d = 3. A negative d makes it decrease, and d = 0 makes every term the same.
What is a geometric sequence?
A geometric sequence multiplies by the same number — the common ratio r — at every step: 3, 6, 12, 24… has r = 2. When |r| < 1 the terms shrink toward zero, which is exactly when an infinite sum exists.
How do I find the nth term of a sequence?
For an arithmetic sequence: aₙ = a₁ + (n − 1)d — with a₁ = 2 and d = 3, the 10th term is 2 + 9 × 3 = 29. For a geometric sequence: aₙ = a₁ · rⁿ⁻¹ — with a₁ = 3 and r = 2, the 10th term is 3 × 2⁹ = 1,536. The calculator shows this working for whatever you enter.
How do I calculate the sum of the first n terms?
Arithmetic: Sₙ = n/2 × (2a₁ + (n − 1)d) — for 2 + 5 + … + 29 (10 terms), that's 5 × 31 = 155. Geometric: Sₙ = a₁(1 − rⁿ)/(1 − r), and if r = 1 simply Sₙ = n · a₁.
What is the sum of an infinite geometric sequence?
When the ratio satisfies |r| < 1, the terms shrink fast enough that the running total settles to a limit: S∞ = a₁/(1 − r). Example: ½ + ¼ + ⅛ + … = 1. When |r| ≥ 1 the series diverges — there is no finite total.
How does the find-the-rule mode work?
The calculator checks your terms for a constant difference (arithmetic), then a constant ratio (geometric), then a constant second difference (quadratic — the rule looks like aₙ = An² + Bn + C), and extends the pattern with predicted terms. Fibonacci-style rules, where each term comes from the two before it, aren't detected yet.
Does the calculator handle fractions, decimals, and negative numbers?
Yes — the first term, the difference or ratio, and the lookup value can all be decimals or negative numbers. Try d = 0.5 or r = −2: the term list, sums, and chart all handle it.
Is the calculator free, and does it store my data?
Free, no sign-up, no limits. Every calculation happens locally in your browser — nothing you type is uploaded, logged, or stored anywhere.