Calculator inputs
Example data table
Sample rule: an = 2an-1 + 3, with a1 = 1.
| n | an |
|---|---|
| 1 | 1 |
| 2 | 5 |
| 3 | 13 |
| 4 | 29 |
| 5 | 61 |
| 6 | 125 |
Formula used
General recursive rule
For a first-order linear recursive sequence, use an = r an-1 + d, with a chosen starting term an₀.
Arithmetic special case
When r = 1, the recursion becomes an = an-1 + d. Then the closed form is an = an₀ + (n - n₀)d.
Geometric special case
When d = 0, the recursion becomes an = r an-1. Then the closed form is an = an₀rn-n₀.
Affine closed form
When r ≠ 1, the closed form is an = rn-n₀(an₀ - c) + c, where c = -d / (r - 1).
How to use this calculator
- Enter the starting index for the sequence.
- Type the seed value at that index.
- Enter the multiplier r for the recursive rule.
- Enter the constant term d added each step.
- Choose how many terms should appear in the table.
- Set the target index for the requested term.
- Select the decimal precision for displayed values.
- Press Calculate Sequence to generate the summary, graph, and exportable table.
Frequently asked questions
1. What is a recursive sequence?
A recursive sequence defines each term from earlier terms. You need an initial value and a recurrence rule. The calculator applies that rule step by step to generate later terms and organize them in notation, table, and graph form.
2. What does function notation mean here?
It means writing the sequence in symbolic form, such as an = r an-1 + d. The page displays the rule, the starting condition, and a matching closed-form expression when the pattern supports one.
3. Can this calculator handle arithmetic sequences?
Yes. Set the multiplier to 1. Then the recursion becomes repeated addition by the constant term. The result behaves like an arithmetic sequence, and the calculator also shows the matching direct formula.
4. Can this calculator handle geometric sequences?
Yes. Set the constant term to 0. Then each term is the previous term multiplied by the chosen ratio. The table and graph will show the geometric growth or decay pattern across the selected indices.
5. Why does the target term differ from displayed terms?
The target index can be larger than the displayed table size. In that case, the calculator keeps iterating internally until it reaches the requested index, then shows the target value separately in the summary cards.
6. What does the graph show?
The graph plots the displayed term values against their indices. It helps you inspect direction, growth speed, curvature, and whether values stabilize, increase, decrease, or alternate over the selected range.
7. When should I export CSV or PDF?
Use CSV when you want spreadsheet analysis, charting, or data sharing. Use PDF when you want a printable report containing the main summary, notation details, and the generated sequence table.
8. What inputs can cause unrealistic results?
Very large multipliers, long target ranges, or unstable starting values can create huge outputs quickly. That may reflect real recursive growth, but it can also cause overflow. Use smaller values for easier interpretation and cleaner plots.