Enter known terms and reveal likely formulas quickly. Check ratios, differences, and missing values instantly. Useful for lessons, exams, research notes, and daily practice.
| Entered terms | Detected pattern | Example rule |
|---|---|---|
| 3, 7, 11, 15, 19 | Arithmetic | a(n) = 3 + (n - 1) × 4 |
| 2, 6, 18, 54, 162 | Geometric | a(n) = 2 × 3^(n - 1) |
| 2, 5, 10, 17, 26 | Quadratic | a(n) = n² + 1 |
| 1, 1, 2, 3, 5, 8 | Fibonacci-like | a(n) = a(n-1) + a(n-2) |
Arithmetic: a(n) = a(s) + (n - s) × d
Geometric: a(n) = a(s) × r^(n - s)
Polynomial: finite differences stay constant at a fixed order
Second-order recurrence: a(n) = p × a(n-1) + q × a(n-2)
This tool returns the simplest matching family it confirms from your entered terms.
A sequence formula finder helps you move from raw terms to a clear mathematical rule. This matters in algebra, number patterns, exam revision, and data interpretation. Students often see a list of values but do not know whether the pattern is arithmetic, geometric, polynomial, or recursive. This calculator speeds up that first check. It compares differences, ratios, and recurrence behavior. It also predicts later terms. That makes practice faster and more structured.
The finder checks common sequence families used in school and college mathematics. Arithmetic sequences have a constant difference. Geometric sequences have a constant ratio. Polynomial sequences show constant finite differences after one or more rounds. Recursive sequences depend on earlier terms. A classic example is a Fibonacci-like rule. Each family tells a different story about growth. Some grow by adding. Some grow by multiplying. Others curve because the change itself changes.
Finite differences are a practical way to recognize polynomial behavior. If first differences are constant, the rule is linear. If second differences are constant, the rule is quadratic. If third differences are constant, the rule is cubic. This method is reliable for evenly spaced term positions. It also helps explain why a sequence bends upward or downward. Instead of guessing, you can inspect the difference table and see the structure directly.
Short sequences can fit more than one valid formula. That is important. For example, four terms may match both a simple pattern and a more complex polynomial rule. The best answer is usually the simplest rule that matches all known terms and the topic you are studying. Add more terms when possible. That reduces ambiguity. This calculator is strongest when you use it as a checking tool, not as a substitute for mathematical reasoning.
Enter at least three terms. Four or more terms usually improve pattern detection. More terms reduce ambiguity and make the returned rule more trustworthy.
Yes. A short list can match many different rules. This finder reports the simplest family it can verify from the entered terms.
It is the n-value linked to your first entered term. If your first value is a(0), use 0. If it is a(1), use 1.
Decimals are supported. The calculator uses numeric comparison with tolerance, so small rounding differences can still be recognized.
The table helps identify linear, quadratic, cubic, and higher polynomial behavior. Constant differences at one level reveal the degree of the pattern.
No. Some sequences need advanced methods, domain knowledge, or many more terms. The tool focuses on common classroom and practical pattern families.
Predicted terms are future values generated from the detected rule. They depend on the rule being correct, so always review the fit first.
Enter more terms when the returned family seems unexpected, when the pattern is irregular, or when several interpretations look possible.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.