# 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.

## Examples

### Recurrence relation example.

The recurrence relation of a sequence is defined by where is a constant with

1. Given that ,  find the value of .
2. Generate the first 4 terms of the sequence.
3. State whether or not the sequence is increasing, decreasing or periodic.

Solution

1. Using the recurrence relation with , . It follows that and so .
2. We already have the first two terms so we need and . Hence, the first 4 terms of the sequence are
3. The sequence is not increasing nor periodic but it is decreasing.

### nth term example

The nth term of a sequence is given by

1. State the period of the sequence.
2. Calculate .

Solution:

1. The period of the sequence is 8.
2. The first 8 terms of the sequence are . Summing these gives a result of zero and since the sequence repeats with period 8, every 8 terms has a sum of 0. It follows that . See more on sigma notation.