Enter sequence details
Use direct mode for known starting values, or derive mode from two known terms.
Example data table
| Sequence type | Known inputs | Explicit formula | Requested result |
|---|---|---|---|
| Arithmetic | a₁ = 4, d = 3, n = 8 | aₙ = 4 + (n - 1) × 3 | a₈ = 25 |
| Arithmetic | a₂ = 7, a₅ = 16 | aₙ = 4 + (n - 1) × 3 | a₁₀ = 31 |
| Geometric | a₁ = 5, r = 2, n = 6 | aₙ = 5 × 2^(n - 1) | a₆ = 160 |
| Geometric | a₂ = 12, a₄ = 48 | aₙ = 6 × 2^(n - 1) | a₇ = 384 |
Formula used
Arithmetic explicit formula: aₙ = aₛ + (n - s)d, where aₛ is the starting term, s is the starting index, and d is the common difference.
Geometric explicit formula: aₙ = aₛr^(n - s), where aₛ is the starting term, s is the starting index, and r is the common ratio.
Arithmetic sum: Sₖ = k/2 × [2aₛ + (k - 1)d].
Geometric sum: Sₖ = aₛ(1 - rᵏ)/(1 - r) when r ≠ 1, and Sₖ = kaₛ when r = 1.
How to use this calculator
- Choose whether your sequence is arithmetic or geometric.
- Select direct mode if you already know the starting term and the common difference or common ratio.
- Select derive mode when you know two terms and need the calculator to infer the explicit rule.
- Enter the starting index, requested term index, and the number of rows for the preview table.
- Press the calculate button to show the explicit formula, nth term, sum, classification, and generated sequence table.
- Use the CSV button for spreadsheets and the PDF button for a shareable printable copy.
Frequently asked questions
1. What is an explicit formula?
An explicit formula gives any term directly from its index. You do not need earlier terms, so it is faster for checking distant values and sums.
2. When should I use arithmetic mode?
Use arithmetic mode when the change between consecutive terms stays constant. Examples include 4, 7, 10, 13 and 20, 15, 10, 5.
3. When should I use geometric mode?
Use geometric mode when each term is found by multiplying by the same ratio. Examples include 3, 6, 12, 24 and 81, 27, 9, 3.
4. What does starting index mean?
Starting index tells the calculator where your starting term lives. Many textbook sequences begin at 1, but some begin at 0 or another integer.
5. Why would I use derive mode?
Derive mode is useful when you know two terms but not the common difference or ratio. It reconstructs the explicit rule from those known positions.
6. Can the calculator handle negative ratios?
Yes, direct geometric mode accepts negative ratios. Derived geometric mode supports real negative ratios when the gap between known term positions is odd.
7. Why is the sum important?
The sum helps when you need totals rather than individual terms. It is useful in finance, coding patterns, growth models, and classroom sequence problems.
8. What do the export buttons do?
CSV export downloads the numeric summary and generated term table. PDF export captures the visible result card and saves it as a portable document.