Recurrence Relations of Sequences

A recurrence relation is another important way to define a sequence. Unlike directly giving the general term formula, a recurrence relation defines the next term through the previous term (or previous few terms). This way of definition is very common in practical problems.

What Is a Recurrence Relation?

A recurrence relation defines later terms through earlier terms.

Examples:

• The Fibonacci sequence: $F_{n+2} = F_{n+1} + F_n$, $F_1 = F_2 = 1$
• An arithmetic sequence: $a_{n+1} = a_n + d$
• A geometric sequence: $a_{n+1} = qa_n$

A recurrence relation must be combined with initial conditions (such as the value of $a_1$) to uniquely determine a sequence.