Calculator Form
Quick examples
Example Data Table
The table below uses a_n = a_{n-1} + a_{n-2} with initial values 0 and 1.
| n | an | Difference | Running Sum |
|---|---|---|---|
| 0 | 0 | — | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 1 | 0 | 2 |
| 3 | 2 | 1 | 4 |
| 4 | 3 | 1 | 7 |
| 5 | 5 | 2 | 12 |
| 6 | 8 | 3 | 20 |
Formula Used
A recursively defined sequence gives each new term from earlier terms. This calculator uses the general model:
a_n = f(n, a_{n-1}, a_{n-2}, a_{n-3}, c, d)
In the form, p1 means the previous term.
p2 means the second previous term.
p3 means the third previous term.
The symbols c and d are optional constants.
The difference is calculated as a_n - a_{n-1}.
The ratio is calculated as a_n / a_{n-1}.
The running sum adds all listed terms up to the current row.
How to Use This Calculator
- Enter the recurrence rule using
p1,p2, orp3. - Enter one to three starting values.
- Set the first index for the first starting value.
- Choose how many terms you want to generate.
- Add constants if your rule uses
cord. - Choose decimal places and an alert threshold.
- Press the calculate button.
- Download the table as CSV or PDF if needed.
Recursive Sequences Explained
What a Recursive Rule Means
A recursive sequence starts with one or more known terms. Each later term is built from earlier terms. This is different from an explicit formula. An explicit formula finds a term directly. A recursive rule grows step by step. That makes it useful for repeated processes.
Why Initial Values Matter
Initial values control the whole sequence. A small change can create a different pattern. Fibonacci style rules need two starting values. Third order rules need three starting values. First order rules usually need only one value. The calculator reads the values in order. The first value belongs to the chosen start index.
Advanced Rule Options
The rule field supports common math functions. You can use square roots, powers, logs, and trigonometry. You can also use constants. This helps model growth, decay, compounding, oscillation, and damping. The variable n lets the rule change by index. Constants c and d make quick testing easier.
Reading the Output
The value column shows each sequence term. The difference column shows step size. The ratio column shows proportional change. The running sum shows accumulated total. Trend notes give a quick behavior check. A threshold alert highlights large terms. This helps find explosive growth early.
Good Use Cases
Recursive sequences appear in finance, biology, computer science, and algebra. They can describe population change. They can model loan balances. They can build numerical approximations. They also help teach patterns. This calculator is useful for lessons, homework checks, and planning tables. Always review the formula before trusting long forecasts.
FAQs
1. What is a recursively defined sequence?
A recursively defined sequence gives new terms from earlier terms. It starts with initial values. Then the recurrence rule generates the rest.
2. What does p1 mean in the formula?
p1 means the previous term. For a term a_n, p1 represents a_{n-1}. It is the most recent sequence value.
3. When should I use p2?
Use p2 when the rule needs the second previous term. Fibonacci style sequences often use p1 plus p2.
4. Can I use powers in the recurrence rule?
Yes. Use the caret symbol or the pow function. For example, p1^2 and pow(p1, 2) both express a square.
5. Why does the ratio show a dash?
The ratio shows a dash when the previous term is zero or missing. Division by zero would not be meaningful.
6. What is the alert threshold?
The alert threshold marks rows where the absolute term value becomes large. It helps identify fast growth or unstable rules.
7. Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a printable report.
8. Is this calculator suitable for nonlinear rules?
Yes. It supports many nonlinear expressions. You can use roots, powers, logarithms, trigonometric functions, and constants.