Definition:Recursive Sequence

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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