Build first and second order recursions with flexible inputs. Generate terms, totals, and comparisons instantly. Save tables for classwork, analysis, reporting, and careful revision.
First order: an = c1an-1 + k
Second order: an = c1an-1 + c2an-2 + k
The calculator starts with the known initial terms. It then applies the selected recurrence rule repeatedly until the requested table length is reached.
Sample rule: an = an-1 + an-2, with a0 = 1 and a1 = 1.
| n | an |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 5 |
| 5 | 8 |
A recursive sequence defines each term from earlier terms. This tool helps you evaluate those patterns quickly. You can model first order and second order rules. You can add a constant term too. The calculator then builds values, shows steps, and summarizes the sequence clearly.
Recursive rules appear in algebra, finance, coding, and science. Population models use them. Loan balances can use them. Signal processing also uses repeated relations. Students often need many terms before noticing a pattern. This page reduces manual work and helps you verify homework or practice tasks with confidence.
The form accepts a starting index, known starting terms, coefficients, and a constant value. You can choose how many terms to generate. You can also request a target index. The result area lists every generated term in order. It also reports the total, average, smallest value, largest value, and net change.
A first order rule uses one previous term. A second order rule uses two previous terms. After you enter the coefficients, each new value is produced from the earlier terms. Because every new output depends on earlier results, a small change in the beginning can strongly affect later values.
Use this page during classwork, revision, or assignment checking. It is useful when tables become long. It is also useful when decimals appear. You can export the term table for records. The printable layout also supports saving the result as a PDF for reference or sharing.
After generating terms, compare the differences between consecutive values. Then compare the ratios when valid. Those checks can reveal whether growth is linear, geometric, alternating, or unstable. Studying both the recurrence rule and the output table gives a better understanding than memorizing formulas alone.
The summary panel gives fast insight. The sum shows overall accumulation. The average shows the central level. The minimum and maximum mark the spread. Net change compares the first and last generated terms. If the growth rate changes sharply, inspect the coefficients and initial values again. That review often explains unexpected behavior in recursive models during real problems.
A recursive sequence defines each term using one or more previous terms. You need initial values before the rule can generate later terms.
A first order rule uses only one earlier term. A second order rule uses two earlier terms. Second order rules can create richer patterns.
The constant term shifts each new value by a fixed amount. It can model regular deposits, losses, offsets, or repeated adjustments.
Yes. The form accepts decimal values for known terms, coefficients, and the constant. You can also choose the displayed decimal precision.
The target index points to one specific term in the sequence. It helps when a question asks for the value at a chosen position.
The ratio compares the last term with the previous term. If the previous term is zero, division is impossible, so the ratio is undefined.
Yes. Use the CSV button to download the generated table. Use the PDF button to print or save the page as a PDF.
Yes. It is useful for checking tables, verifying target terms, comparing patterns, and reviewing how the recurrence rule behaves.
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.