# Definition:Recursive Sequence

(Redirected from Definition:Difference Equation)

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