Calculator Inputs
Example Data Table
| Type | Inputs | Rule | Requested term | Output |
|---|---|---|---|---|
| Arithmetic | a0 = 4, d = 3 | an = an-1 + 3 | a6 | 22 |
| Geometric | a0 = 5, r = 2 | an = 2an-1 | a5 | 160 |
| First-order affine | a0 = 2, p = 1.5, c = 1 | an = 1.5an-1 + 1 | a4 | 16.375 |
| Second-order linear | a0 = 0, a1 = 1, p = 1, q = 1, c = 0 | an = an-1 + an-2 | a7 | 13 |
Formula Used
Arithmetic recursion: an = an-1 + d, with closed form an = a0 + nd.
Geometric recursion: an = r an-1, with closed form an = a0rn.
First-order affine recursion: an = p an-1 + c. When p ≠ 1, an = pna0 + c(pn − 1)/(p − 1).
Second-order linear recursion: an = p an-1 + q an-2 + c. The homogeneous part uses the characteristic equation λ2 − pλ − q = 0. The calculator also finds steady-state behavior when available.
How to Use This Calculator
- Choose the recurrence type that matches your sequence rule.
- Enter the required initial term values and coefficients.
- Set the target index n for the requested term.
- Choose how many generated terms should appear in the results table.
- Pick a precision level for rounded numeric output.
- Press Solve Sequence to display the result above the form.
- Use the export buttons to save a CSV or PDF copy.
FAQs
1) What kinds of sequences can this solver handle?
It handles arithmetic, geometric, first-order affine, and second-order linear recursions. These forms cover many textbook problems, growth models, and classic sequences such as Fibonacci-style recurrences.
2) What does the requested value mean?
The requested value is the term at your chosen index n. If you enter n = 8, the calculator evaluates a8 using the recurrence rule and the starting terms.
3) Why are both a0 and a1 needed sometimes?
Second-order recursions depend on two earlier terms. Because each new term uses an-1 and an-2, two starting values are needed before the rule can continue.
4) Does the solver always show a closed form?
Not always. Some resonance and complex-root cases are better shown numerically. The calculator still generates accurate terms and explains the recurrence structure, even when a compact explicit formula is less practical.
5) What is the purpose of the change and ratio columns?
They help you inspect behavior quickly. The change column highlights additive movement, while the ratio column reveals multiplicative behavior, sign flips, and convergence or divergence patterns.
6) Can I use decimal or negative coefficients?
Yes. The inputs accept integers, decimals, and negative values. That allows you to test damping, oscillation, contraction, alternating signs, and many other useful mathematical cases.
7) Why might I see an overflow warning?
Some recursions grow extremely fast. Large coefficients or a large target index can exceed normal numeric limits. Reduce n or use smaller coefficients if the sequence becomes too large.
8) What is included in the CSV and PDF exports?
The exports include the recurrence summary, the requested term, key metrics, and the generated table of terms. This makes it easier to document classwork, share results, or save study notes.