Free number sequence calculator
Generate arithmetic or geometric number sequences — enter the starting value, common difference or ratio, and number of terms to see the full sequence, updated live, as you type.
On this page10 sections
Arithmetic: each term += common difference. Geometric: each term ×= common ratio.
Results are estimates. Consult a professional.
How the number sequence calculator works
A number sequence is an ordered list of numbers following a rule. The two most important types are arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms). Given the first term, the common difference or ratio, and the position n, the calculator finds any term and optionally the sum of the first n terms.
Worked example: arithmetic and geometric sequences
Find the first five terms and the sum of the first 5 terms for an arithmetic sequence starting at 2 with common difference 3, and a geometric sequence starting at 3 with ratio 2.
First 10 terms of common sequences
These four sequences appear throughout mathematics, computer science, and nature. Knowing their first terms helps you identify patterns and verify calculations quickly.
| Sequence | Rule | First 10 terms |
|---|---|---|
| Natural numbers | n | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 |
| Perfect squares | n² | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| Powers of 2 | 2^n (n≥0) | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 |
| Fibonacci | F(n)=F(n-1)+F(n-2) | 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 |
| Triangular numbers | n(n+1)/2 | 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 |
| Odd numbers | 2n − 1 | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 |
Fibonacci starts F(1)=1, F(2)=1. Powers of 2 start at 2⁰=1.
Tips for number sequences
Identifying the sequence type is the first step — once you know whether it is arithmetic or geometric, the formulas do the rest.
- Test for arithmetic first — Subtract consecutive terms. If the differences are all equal, it is arithmetic. The common difference d can be negative (decreasing sequence) or fractional.
- Test for geometric next — Divide consecutive terms. If the ratios are all equal, it is geometric. A ratio between 0 and 1 produces a decreasing sequence; a negative ratio alternates signs.
- Sum formula shortcut (arithmetic) — Gauss's trick: pair the first and last term, multiply by the count, divide by 2. S₁₀₀ of natural numbers = 100 × 101 / 2 = 5,050.
- Infinite geometric series — When |r| < 1, the infinite sum converges: S∞ = a₁ / (1 − r). For example, 1 + ½ + ¼ + … = 1 / (1 − 0.5) = 2.
- Recursive vs. explicit formulas — The explicit formula (aₙ = …) lets you jump straight to any term. The recursive definition (aₙ = aₙ₋₁ + d) requires computing every previous term first.
Accuracy and limitations
The arithmetic and geometric formulas are algebraically exact. For very large n or very large r, geometric sequence terms grow exponentially and may exceed the range of standard floating-point numbers (approximately 1.8 × 10³⁰⁸ for 64-bit IEEE 754). Fibonacci and other recursive sequences can be computed exactly for moderate n; for very large n (beyond roughly 75 terms), values exceed JavaScript's safe integer range and require arbitrary-precision arithmetic.
Key terms
About this calculator
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