Sequence Calculator
Analyze any number sequence to find the formula, next terms, and sum. Our tool automatically detects **Arithmetic**, **Geometric**, and **Quadratic** sequences.
Enter at least 3 numbers, separated by commas or spaces.
How to Find the Rule for a Sequence
This calculator is a powerful tool that automatically checks for the most common types of mathematical sequences. Here are the formulas it uses:
1. Arithmetic Sequence
A sequence is **arithmetic** if the difference between terms is constant. This is called the "common difference" (d).
- Example: 5, 8, 11, 14... (The common difference `d` is 3).
- Explicit Formula (nth term): `aₙ = a₁ + (n-1)d`
- Sum Formula (Series): `Sₙ = n/2 * (2a₁ + (n-1)d)`
2. Geometric Sequence
A sequence is **geometric** if the ratio between terms is constant. This is called the "common ratio" (r).
- Example: 3, 6, 12, 24... (The common ratio `r` is 2).
- Explicit Formula (nth term): `aₙ = a₁ * rⁿ⁻¹`
- Sum Formula (Series): `Sₙ = a₁ * (1 - rⁿ) / (1 - r)`
3. Quadratic Sequence
A sequence is **quadratic** if the *second* difference (the difference between the differences) is constant. The formula is in the form of a quadratic equation.
- Example: 1, 4, 9, 16... (The sequence of perfect squares).
- Explicit Formula (nth term): `aₙ = An² + Bn + C`
Frequently Asked Questions (FAQ)
What is the difference between an arithmetic and geometric sequence?
An **arithmetic** sequence has a *common difference* (you add or subtract the same number each time). A **geometric** sequence has a *common ratio* (you multiply or divide by the same number each time).
Is the Fibonacci sequence (1, 1, 2, 3, 5) arithmetic or geometric?
Neither. The Fibonacci sequence is a **recursive** sequence, where the next term is the sum of the two preceding ones. It does not have a constant difference (1-1=0, 2-1=1) or a constant ratio (2/1=2, 3/2=1.5). Our calculator does not currently detect recursive sequences like Fibonacci.
Formula (LaTeX) + variables + units
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- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.