Find Sequence Formula Calculator

Calculator

Enter terms separated by commas, spaces, or new lines. Fractions like 3/4 are allowed.

Example data table

These are common patterns the calculator can identify.

Sequence	Detected type	Example nth-term rule	Next term
2, 5, 8, 11, 14	Arithmetic	aₙ = 3n - 1	17
3, 6, 12, 24, 48	Geometric	aₙ = 3 · 2^(n-1)	96
1, 4, 9, 16, 25	Quadratic	aₙ = n²	36

Formula used

Arithmetic sequences

Constant first difference: d = aₙ − aₙ₋₁

aₙ = aₖ + (n − k)·d

Here, k is your chosen start index.

Geometric sequences

Constant ratio: r = aₙ / aₙ₋₁

aₙ = aₖ · r^(n − k)

Works best when terms are non‑zero and stable.

Quadratic sequences

Constant second difference: Δ²a

aₙ = an² + bn + c

Coefficients are solved from the first three terms.

How to use this calculator

Paste your sequence terms in the input box.
Choose whether indexing starts at n=1 or n=0.
Set how many future terms you want to predict.
Press Submit to see the detected model and formula.
Use CSV or PDF export for quick sharing.

Pattern coverage metrics

With 3 to 8 terms, the calculator targets three high-frequency families: arithmetic, geometric, and quadratic. Arithmetic detection checks constant first differences across all adjacent pairs. Geometric detection checks a stable ratio, avoiding division by values near zero. Quadratic detection verifies constant second differences, then solves a, b, and c.

Finite differences as evidence

A finite-differences table compresses a sequence into trend layers. Level 1 highlights additive growth; a constant row implies an arithmetic rule. Level 2 highlights curvature; a constant row implies a quadratic rule. This page prints up to three levels so you can validate the model visually, not just accept a label. For example, the sequence 1, 4, 9, 16 shows first differences 3, 5, 7 and second differences 2, 2. That constant 2 is twice the quadratic coefficient a. The chart overlays your fitted rule and reveals whether residuals drift upward or oscillate. Use it to spot breaks after a specific term or index very quickly.

Indexing and formula alignment

Many textbooks define the first term at n=1, while programming tasks often start at n=0. The Index Start control shifts the derived rule so that your first entered value aligns with the chosen n. This reduces off-by-one errors when you reuse the formula in homework, reports, or scripts.

Prediction window control

The Predict Next Terms input caps forecasts at 20 values to keep results readable and exports compact. For arithmetic, each step adds d. For geometric, each step multiplies by r. For quadratic, each step evaluates an·n² + bn + c at successive indices. The table lists both n and value for traceability.

Tolerance for real-world decimals

Measured data can introduce rounding, sampling noise, or unit conversions. The Tolerance setting relaxes equality checks using a relative threshold. If a sequence almost matches but fails, increase tolerance gradually, for example from 1e-9 to 1e-6. A larger tolerance may fit noise, but it can also mask genuine pattern changes.

Exports and reproducibility

CSV export captures the detected type, formula, parameters, input terms, predicted terms, and the generation timestamp. PDF export produces a clean, shareable summary suitable for submissions. Together they support reproducibility: a reviewer can rerun the same inputs and confirm the same rule and forecasts.

FAQs

1) Why does the calculator need at least three terms?

Three terms allow reliable checks for arithmetic and geometric rules and enable solving a quadratic model. More terms strengthen validation and reduce the chance of matching a coincidental pattern.

2) What does the tolerance setting change?

Tolerance controls how strictly differences or ratios must match. Increase it when decimals include rounding noise. Keep it small for exact integer sequences to avoid accepting an incorrect model.

3) When will geometric detection fail?

Geometric fitting may fail when a term is zero or extremely close to zero, because ratios become unstable. In that case, consider using more terms or analyzing differences instead.

4) What does “index starts at n=0” mean?

It means your first entered term is treated as a₀ rather than a₁. The displayed formula shifts accordingly, which is useful for programming tasks and recurrence-based definitions.

5) Can this find formulas for every possible sequence?

No. It focuses on common closed forms: arithmetic, geometric, and quadratic. Sequences with mixed rules, factorial growth, or alternating patterns may not be confidently classified.

6) What is included in the CSV and PDF exports?

Exports include the detected type, the nth-term formula, key parameters, your input terms, predicted terms, and a timestamp. This helps you share results and reproduce the same run later.