Sequence inputs and options
Quick examples
Formula used
An arithmetic sequence uses aₙ = a₁ + (n − 1)d. Its finite sum is Sₙ = n[2a₁ + (n − 1)d]/2. A geometric sequence uses aₙ = a₁rⁿ⁻¹.
A finite geometric sum uses Sₙ = a₁(1 − rⁿ)/(1 − r). When |r| < 1, the infinite sum is S∞ = a₁/(1 − r). Polynomial patterns are identified through constant finite differences.
How to use this calculator
Enter terms in their original order. Replace unknown values with a question mark. Select automatic detection unless you already know the pattern.
Choose future terms, precision, sums, and target positions. Add another sequence when comparison is useful. Press the calculation button to view every result.
How pattern detection works
The calculator tests constant differences and ratios first. It then checks harmonic, recursive, alternating, square, triangular, factorial, and prime rules. Higher differences reveal polynomial degree.
Short sequences remain mathematically ambiguous. Several formulas can reproduce the same finite list. The alternatives section therefore shows other exact matches.
Explicit and recursive formulas
An explicit formula calculates a term directly from its position. A recursive formula depends on one or more earlier terms. Both descriptions can represent the same sequence.
Custom formulas use a safe expression parser. Explicit rules use the variable n. Recursive rules use a or prev.
Worked examples
| Sequence | Likely pattern | Next terms |
|---|---|---|
| 2, 5, 8, 11 | Add 3 | 14, 17, 20 |
| 3, 6, 12, 24 | Multiply by 2 | 48, 96, 192 |
| 1, 4, 9, 16 | Consecutive squares | 25, 36, 49 |
| 2, 6, 12, 20 | Quadratic differences | 30, 42, 56 |
| 1, 1, 2, 3, 5 | Previous two terms added | 8, 13, 21 |
Common pattern-finding mistakes
Do not assume the simplest visible operation is unique. Check alternating positions and higher differences. Confirm a rule against every supplied term.
Geometric ratios need special care around zero. Fractions may look irregular after rounding. Increase precision before rejecting a likely match.
Frequently asked questions
Can the calculator fill missing terms?
Yes. Use question marks or x characters. The selected model estimates those positions.
Why are several rules shown?
Any finite list can fit many formulas. Alternative matches help expose that ambiguity.
Does it support fractions?
Yes. Enter fractions such as 3/4. Results can also display as fractions.
What is a difference table?
It repeatedly subtracts neighboring terms. A constant row indicates polynomial degree.
Can I calculate partial sums?
Yes. Choose the required number of terms. The result includes their total.
Can I find a term position?
Yes. Enter a target value. The calculator searches generated terms for its first match.
Are custom formulas evaluated safely?
The included parser accepts only supported numbers, variables, operators, and mathematical functions.
How accurate is polynomial interpolation?
It exactly fits supplied numeric points, within floating-point precision. Future behavior may still be uncertain.
Can I compare two sequences?
Yes. Enter the optional second sequence. The results compare values and position differences.
Does the calculator prove convergence?
It reports useful observations and geometric infinite sums. Formal convergence proofs require additional analysis.
Can results be exported?
Yes. Use CSV for data. Use printing to save a PDF copy.