Model recurrence rules, inspect terms, and compare patterns. Use flexible inputs for practical math exploration. Export clean outputs for classes, homework, revision, and review.
A linear recursive sequence follows a fixed update rule. The general form is shown below.
a(n) = c1·a(n-1) + c2·a(n-2) + ... + ck·a(n-k) + d
Here, c1 to ck are the coefficients. The value d is an optional constant term. The first k sequence values are the initial terms. After that, every new term is computed from earlier terms.
For homogeneous cases, set d = 0. The calculator also shows the characteristic polynomial for the homogeneous part. That polynomial helps with recurrence analysis, stability study, and pattern recognition.
This sample uses the recurrence a(n) = a(n-1) + a(n-2) with initial terms 0, 1.
| Index | Term |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
Linear recursive sequences appear throughout mathematics. They also appear in algorithms, coding theory, finance, and discrete models. This calculator helps you compute terms quickly and correctly. You can enter any order of recurrence. You can also add a constant forcing term. That makes the tool useful for both homogeneous and non-homogeneous patterns. Students often need the nth term, a table of values, and a clean rule display. This page gives all three. It also shows summary values for the generated terms. That helps you inspect growth, spread, and behavior.
A recurrence relation defines each new term from earlier terms. The coefficients control how strongly previous values affect the next result. The initial terms act like the starting engine of the sequence. Small changes there can create very different outputs. This is why recursive sequence analysis matters in proofs and modeling. The characteristic polynomial is also important. It connects the recurrence to algebraic structure. In many courses, it helps explain long-run growth, oscillation, and repeated roots. Even when you only need a table, understanding the rule improves accuracy.
The result section is designed for quick reading. You see the recurrence rule first. You also see the characteristic polynomial. Then the calculator highlights the order, target index, target term, and displayed length. The output table lists each index and term. It also compares each value with the previous one. That ratio column helps when the sequence grows steadily. If a previous term is zero, the ratio is left blank. The summary boxes add the sum, average, minimum, and maximum of the displayed terms. This supports classwork, homework, and verification.
Start with a simple known recurrence. Check the output table against manual work. Then move to higher-order rules. Try negative coefficients to study alternating behavior. Try a constant term to inspect shifted growth. You can also apply a modulus for modular sequence tasks. That is useful in number theory and programming practice. The export buttons help you save results for reports or revision sheets. Because the layout stays clean and direct, the calculator works well for quick problem solving and deeper recurrence exploration.
It is a sequence where each new term is built from earlier terms using fixed coefficients. Many textbook sequences use this structure because it is systematic and easy to analyze.
The order of the recurrence decides how many previous terms are needed. You must provide exactly that many starting values so the calculator can generate the next terms correctly.
The constant term adds the same fixed amount at each recursive step. It turns a homogeneous recurrence into a non-homogeneous one and can noticeably change growth behavior.
It summarizes the homogeneous part of the recurrence in algebraic form. It is useful for studying roots, repeated factors, long-run behavior, and closed-form derivations.
Yes. The calculator accepts decimal coefficients, decimal initial terms, and a decimal constant term. Precision settings control how many decimals are shown in the output.
The calculator reduces each newly generated term by the modulus. This is useful for modular arithmetic tasks, periodic pattern checks, and programming-related recurrence problems.
The ratio uses the previous term as the divisor. If that earlier term is zero, the ratio is undefined, so the calculator leaves that entry empty.
Yes. You can export a CSV file for spreadsheets or a PDF file for sharing, printing, or attaching to homework and study notes.
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.