Sequence and Series > Periodic sequence

## What is a Periodic Sequence?

A **periodic sequence** (also called a *cycle* or *train*) is a sequence that repeats itself. It is a special type of periodic function if its domain is the set of natural numbers (n = 1, 2, …).

More formally, the definition is:

**a _{i + np} = a_{i}. **

Every periodic sequence has a **period**, N (the length of the period or how many terms are in the repetition) so that [1]

**x(n) = x(n + N) for all n. **

As an example, the sequence {-1, 1, -1, 1,…} has period 2 and {1, 1, 1, 1, …} has period 1.

A **psuedo-periodic sequence** behaves in a similar way, but not quite [2]. For example, the sequence:

{1, 2, 3, 2, 3, 4, 3, 4, 5…}

starts with 1, 2, 3. The next three terms are 1(+1), 2(+1), 3(+1), and so on: adding 1 to each string of three in the sequence.

## Examples of a Periodic Sequence

- {1, 2, 1, 2, 1, 2,…}
- The decimal expansion of 1/13 = 0.0 769230 769230…
- Periodic Binary Sequences (binary numbers that repeat): {1, 0, 1, 0, 1, 0,…) or {1, 1, 0, 1, 1, 0,…)

## Purely and Ultimately Periodic

A **purely periodic sequence** has all repeating terms. For example, in the sequence {1, 2, 3, 1, 2, 3, 1, 2, 3} the numbers {1, 2, 3} repeat.

If the numbers eventually get to repeat (i.e. it’s periodic from some point onwards), then the sequence is ultimately periodic (also called *aperiodic* or *eventually periodic*). For example, the **ultimately periodic sequence** {1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, …} starts off as non-periodic but eventually repeats {1, 2, 4} to infinity.

An example of a function that gives an eventually periodic sequence is

**F = f ^{1}(n), f^{1}(n),… f^{k}(n), **

Where f^{k}(n) is defined as f^{1}(n) = f(n) and f^{k+1}(n) = f(f^{k}(n)).

If n is a non-negative integer then the sequence is eventually periodic starting at the 4th term ( 1, 2, 7, 5, 4, 6, 5, 4, 6, 5, 4, 6) [3].

A great trick to finding out if a sequence is ultimately periodic is to create a bar graph from the list of terms in the sequence. A visual check may reveal a pattern.

## References

[1] Frequency Analysis of Discrete Time Signals. Retrieved April 7, 2021 from: https://engineering.purdue.edu/~ee538/Chap4_DFTsinewaves.pdf

[2] MacGregor, R. Generalizing the Notion of a Periodic Sequence. Retrieved April 7, 2021 from: https://www.jstor.org/stable/2321983?seq=1

[3] Problem C: Eventually periodic sequence. Retrieved April 7, 2021 from: https://web.eecs.umich.edu/~kjc/acm/oct27/C.pdf