# Definition:Recursive Sequence

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Definition:Recursively Defined Mapping.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## Definition

A **recursive sequence** is a sequence where each term is defined from earlier terms in the sequence.

A famous example of a recursive sequence is the Fibonacci sequence:

- $F_n = F_{n-1} + F_{n-2}$

The equation which defines this sequence is called a **recurrence relation** or **difference equation**.

### Initial Terms

Let $S$ be a recursive sequence.

In order for $S$ to be defined, it is necessary to define the **initial term** (or terms) explicitly.

For example, in the Fibonacci sequence, the **initial terms** are defined as:

- $F_0 = 0, F_1 = 1$

## Also see

- Inductive Definition of Sequence for the justification of this definition.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**difference equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**difference equation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**difference equation**