Recursive Sequence Calculator
Generate terms from aₙ = r·aₙ₋₁ + d, including Fibonacci-style rules.
Enter your values
Enter the first term a₁ you are working with.
Enter the multiplier r you are working with.
Enter the added constant d you are working with.
Enter the terms to generate you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- nth term
- 301.546875
- First terms
- 2, 6, 12, 21, 34.5, 54.75, 85.125, 130.688, 199.031, 301.547
- Sum of all terms
- 846.640625
- Long-run limit
- Diverges (|r| ≥ 1)
- Behaviour
- Growing without bound
Quick answer
A linear recurrence aₙ = r·aₙ₋₁ + d builds each term from the last; it converges to d ÷ (1 − r) when |r| < 1.
aₙ = r·aₙ₋₁ + d
At a glance
| What it does | Generate terms from aₙ = r·aₙ₋₁ + d, including Fibonacci-style rules. |
|---|---|
| Category | Math |
| Inputs needed | First term a₁, Multiplier r, Added constant d, Terms to generate |
| Main output | nth term |
| Formula | aₙ = r·aₙ₋₁ + d |
| Cost | Free — no sign-up, no download |
How to use the Recursive Sequence Calculator
- 1Enter the starting term.
- 2Enter the multiplier and the constant added each step.
- 3Set how many terms you want listed — the nth term and running total appear instantly.
Inputs explained
- First term a₁
- Enter the first term a₁ you are working with.
- Multiplier r
- Enter the multiplier r you are working with.
- Added constant d
- Enter the added constant d you are working with.
- Terms to generate
- Enter the terms to generate you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- First term a₁2
- Multiplier r1.5
- Added constant d3
- Terms to generate10
Results
- nth term301.546875
- First terms2, 6, 12, 21, 34.5, 54.75, 85.125, 130.688, 199.031, 301.547
- Sum of all terms846.640625
- Long-run limitDiverges (|r| ≥ 1)
- BehaviourGrowing without bound
Frequently asked questions
Can it do Fibonacci?
Fibonacci needs the two previous terms, so it is a second-order recurrence. This tool covers first-order rules of the form aₙ = r·aₙ₋₁ + d.
What is the fixed point?
The value where the sequence stops changing: d ÷ (1 − r), which the sequence approaches when |r| < 1.
What do I need to enter into the Recursive Sequence Calculator?
Just 4 values: first term a₁, multiplier r, added constant d, terms to generate. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Recursive Sequence Calculator work out the answer?
A linear recurrence aₙ = r·aₙ₋₁ + d builds each term from the last; it converges to d ÷ (1 − r) when |r| < 1. It applies the formula aₙ = r·aₙ₋₁ + d and shows the working so you can check each step by hand.
Is the Recursive Sequence Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.