Build terms from common rules easily quickly. Compare sequence models, preview tables, and chart values. Solve nth terms, sums, and patterns with confident accuracy.
Arithmetic: aₙ = a₁ + (n - 1)d
Geometric: aₙ = a₁ × r^(n - 1)
Quadratic: aₙ = a₁ + (n - 1)Δa₁ + ((n - 1)(n - 2)/2)Δ²a
Recurrence: aₙ = p·aₙ₋₁ + q·aₙ₋₂
Arithmetic sequences move by a fixed step. If d = 5 and a₁ = 2, the terms are 2, 7, 12, 17, and so on.
Geometric sequences scale by a fixed multiplier. If r = 3 and a₁ = 2, the terms are 2, 6, 18, 54, and so on.
Quadratic sequences keep the second difference constant. They are useful when the change between terms grows linearly.
Second-order recurrence sequences create each new term from the prior two terms. Fibonacci-style patterns are a famous example.
| Sequence Type | Input Example | First 6 Terms | Example Target Result |
|---|---|---|---|
| Arithmetic | a₁ = 4, d = 3 | 4, 7, 10, 13, 16, 19 | a₁₀ = 31 |
| Geometric | a₁ = 2, r = 2 | 2, 4, 8, 16, 32, 64 | a₈ = 256 |
| Quadratic | a₁ = 1, Δa₁ = 3, Δ²a = 2 | 1, 4, 9, 16, 25, 36 | a₇ = 49 |
| Recurrence | a₁ = 1, a₂ = 1, p = 1, q = 1 | 1, 1, 2, 3, 5, 8 | a₁₀ = 55 |
It supports arithmetic, geometric, quadratic, and second-order recurrence sequences. Each model uses its own rule inputs, then produces terms, a target n-th value, summary metrics, and a graph.
A quadratic sequence has a constant second difference. The gap between consecutive terms changes by the same amount each step, which often matches square-number and parabola-related patterns.
The calculator shows only fields needed for the selected sequence type. This keeps the form cleaner and reduces entry mistakes for rules that do not use certain parameters.
Yes. The displayed table can show fewer terms, while the target n-th term can be larger. The calculator internally generates enough values to return the requested index.
The recurrence model starts with a₁ and a₂. Every later term uses the formula aₙ = p·aₙ₋₁ + q·aₙ₋₂, letting you model Fibonacci-style and custom recursive patterns.
It compares the first displayed term with the last displayed term. This helps you quickly see whether the sequence is increasing, decreasing, or changing sharply over the shown range.
Very large ratios, coefficients, or target indices can create values beyond normal numeric limits. Reducing the ratio, coefficients, or total number of generated terms usually resolves this.
CSV exports the generated term table for spreadsheet use. PDF exports the summary details and table, making it easier to save results for notes, homework, or reports.
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.