Recursive Sequence Calculator

Q: Starting at 2 with aₙ₊₁ = 3aₙ + 1?

2, 7, 22, 67, 202, 607, 1822, 5467.

Q: When does such a sequence converge?

When |m| < 1, in which case it approaches the fixed point regardless of the starting value.

Work out recursive sequence instantly with clear inputs, formula shown and shareable results.

About Recursive Sequence Calculator

A first-order linear recurrence aₙ₊₁ = m·aₙ + c generates each term from the previous one. Its fixed point c/(1 − m) is the value that maps to itself, and the sequence converges to it when |m| < 1.

Formula

Linear recurrence

aₙ₊₁ = m·aₙ + c, fixed point L = c / (1 − m)

Worked example

With the default values below, Recursive Sequence Calculator returns Last term generated of 5,467.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

First term: 2
Multiply the previous term by: 3
Then add: 1
How many termsaccepts 1–60: 8

Results

Last term generated: 5,467.00000000
Sequence: 2, 7, 22, 67, 202, 607, 1822, 5467
Terms generated: 8
Fixed point c/(1 − m): -0.50000000

Step by step

1Values used
2Linear recurrence
3Last term generated= 5,467.00000000
4Sequence= 2, 7, 22, 67, 202, 607, 1822, 5467
5Terms generated= 8
6Fixed point c/(1 − m)= -0.50000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Starting at 2 with aₙ₊₁ = 3aₙ + 1?