Number Sequence Calculator
Enter a sequence and find the pattern — predict the next terms automatically.
Results
How to Identify Number Sequence Patterns
A number sequence is an ordered list of numbers following a pattern or rule. Identifying the pattern allows you to predict any future term. The most common types are arithmetic, geometric, and polynomial (square, cube) sequences.
Arithmetic Sequences
Constant first differences between consecutive terms. Example: 3, 7, 11, 15... — differences are always 4. Pattern: add 4 each time. Next term: 19, 23, 27...
Geometric Sequences
Constant ratio between consecutive terms. Example: 2, 6, 18, 54... — ratio is always 3. Next term: 162, 486...
Square and Cube Sequences
1, 4, 9, 16, 25... are perfect squares (n²). Their first differences (3, 5, 7, 9...) are arithmetic, and second differences are constant (2). Cube sequence: 1, 8, 27, 64, 125... (n³).
Fibonacci-like Sequences
Each term is the sum of the two preceding terms. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13... is the most famous example. Any sequence where each term = sum of previous two follows this pattern.
How Pattern Detection Works
Check differences (first, second, third) and ratios between terms. Constant first difference = arithmetic. Constant ratio = geometric. Constant second difference = quadratic (n²). This calculator analyzes differences and ratios to identify the most likely pattern.
Recognizing Sequence Types: A Worked Example
Given: 2, 5, 10, 17, 26... How do you identify this?
First differences: 3, 5, 7, 9 — not constant, so not arithmetic.
Second differences: 2, 2, 2 — constant! This means the sequence is quadratic.
Pattern: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26. Formula: aₙ = n²+1. Next term: 6²+1 = 37.
Common Sequence Types Quick Reference
| Sequence | Example | Pattern | Next Term |
|---|---|---|---|
| Arithmetic | 3, 7, 11, 15 | +4 each time | 19 |
| Geometric | 2, 6, 18, 54 | ×3 each time | 162 |
| Perfect squares | 1, 4, 9, 16, 25 | n² | 36 |
| Fibonacci-like | 1, 1, 2, 3, 5, 8 | Sum of two previous | 13 |
| Triangular | 1, 3, 6, 10, 15 | +1, +2, +3... | 21 |
Frequently Asked Questions
Compute consecutive differences (first differences). If constant: arithmetic. If not, check ratios between terms. If constant: geometric. If second differences are constant: quadratic.
64. Each term doubles — this is a geometric sequence with r=2.
36. These are perfect squares (1²,2²,3²,4²,5²,6²).
13. Fibonacci sequence: each term = sum of two preceding terms.
No. Some sequences are random or defined by complex formulas. Pattern detection works well for arithmetic, geometric, and polynomial sequences.
Triangular numbers are 1, 3, 6, 10, 15, 21... — the n-th triangular number is n(n+1)/2. They count the number of objects that can be arranged in an equilateral triangle. Their first differences are 2, 3, 4, 5... (arithmetic), and their second differences are all 1. The sum of the first n integers equals the n-th triangular number.
If second differences are constant (call it 2a), then the sequence formula is aₙ = an² + bn + c. Find a from the second difference (a = second_diff/2), then substitute known terms to solve for b and c. For 3, 8, 15, 24...: second differences = 2, so a=1. Substituting n=1: 1+b+c=3, and n=2: 4+2b+c=8. Solving gives b=2, c=0, so aₙ = n²+2n.