# Recurrence Relations

We often refer to sequences defined by recurrence relations as term-to-term sequences. Recall that is the th term in a given sequence. A recurrence relation is defined as follows:

.

As you can see, the next term in a sequence is a function of the previous term. In order to generate a sequence like this, the first term must be stated. For example, consider the sequence generated by:

We find by setting and substituting . We find by setting and substituted and so on. Hence why it is called a ‘recurrence relation’. This sequence is nonlinear and the terms are given by

,

,…

… and so on. After the first term, the terms in this sequence will always be 12. See Example 1 for another recurrence relation.

## nth term sequences

We often refer to th term sequences as position-to-term sequences. This is where the sequences are defined as follows:

This is also known as closed form. In this case, each term in the sequence if a function of its position. Unlike for recurrence relations, we do not need a first term to generate a sequence like this. This is because, when given in closed form, each term of a sequence is defined by its position number. For example, consider the sequence generated by the closed form . We find by setting , by setting etc:

,

,

, …

and so on. This sequence is also nonlinear and will diverge (what does diverge mean?). See Example 2 for more.

## Increasing, Decreasing and Periodic Sequences

- Increasing sequences satisfy . For example, the sequence generated by the recurrence relation with is given by This is an increasing sequence.
- Decreasing sequences satisfy . For example, the sequence generated by the th term rule is given by . This is a decreasing sequence.
- Periodic sequences are those that repeat after a fixed number of terms. We can write this as for some fixed value of . is known as the order or period of the sequence. For example, the sequence is periodic with period 3.

See Examples 1 and 2.