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What is a periodic sequence?

A period sequence is a sequence with a repeating pattern of terms. The following sequences are all periodic:

• $0, 1, 0, 1, 0, 1, ...$
• $3, 1, 4, 1, 5, 9, 3, 1, 4, 1, 5, 9, ...$
• $e, \pi, 3, e, \pi, 3, e, \pi, 3, ...$
• $7, 7, 7, 7, 7, 7, ...$

What is a period of a sequence?

The period of a sequence is the number of terms within the repeated part of a sequence. This will always be a positive whole number. Let's look at the periods of the aforementioned sequences:

• $\boxed{0, 1,} 0, 1, 0, 1, ...$ has period $2$.
• $\boxed{3, 1, 4, 1, 5, 9,} 3, 1, 4, 1, 5, 9, ...$ has period $6$.
• $\boxed{e, \pi, 3,} e, \pi, 3, e, \pi, 3, ...$ has period $3$.
• $\boxed{7,} 7, 7, 7, 7, 7, ...$ has period $1$.

Periodic sequences and rational numbers!

The decimal expansions of rational numbers always have periodic sequences! Let's look at a few examples:

• $\frac{1}{3}=0.333333333...$ has period $1$.
• $\frac{74}{999}=0.074074074074...$ has period $3$.
• $\frac{5}{7}=0.71428571428...$ has period $6$.

This property is true for rational numbers whose decimal expansions are finite as well! Technically, any finite decimal can be followed by infinitely many zeroes:

• $\frac{3}{4}=0.75=0.75000000...$ has a sequence with period $1$ (i.e. the infinite sequence of $0$s at the end of the fraction).

In Real Life

If you're reading the hours on the clock, it goes in the following sequence: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, ...$. Because this repeats forever, we know the sequence of hours on a clock is periodic! The same thing occurs for seconds in a minute, minutes in an hour and days in a year. There are many examples of these kinds of sequences but a clock is a simple one to think about.

Periodic sequences

Periodic sequences continually repeat themselves every $p$ terms. A periodic sequence can be thought of as the discrete version of a periodic function. In particular, for a periodic sequence $\{a_n\}$, there exists a positive integer constant $p$ such that for all $n$ in thhe natural numbers, $$a_n=a_{n+p}.$$ The constant $p$ is said to be the period of the sequence. The lowest positive constant $p$ possible is called the fundamental period of the sequence.

As another explicit example, the sequence $$1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, \ldots$$ is periodic because the numbers $1, 4, 2, 8, 5, 7$ repeat themselves over and over, in that order, every $6$ terms. So, in this sequence, the fundamental period of the sequence is $6.$

You may have noticed in your journey with the decimal representations of certain fractions, that you have bumped into periodic sequences! For example, consider the decimal representation of $\dfrac{1}{7}$: $$\dfrac{1}{7}=0.1428571428\dots$$ Each digit corresponds to the next term of a periodic sequence! As another example, $$\dfrac{1}{99}=0.010101\dots$$ Do all rational numbers $\dfrac{p}{q}$ satisfy that their decimal representations are periodic sequences? Unfortunately not; for example, $$\dfrac{1}{6}=0.16666\dots$$ and the starting value $a_1=1$ does not repeat at all. However, it is a true fact that the decimal representations of all rational numbers $\dfrac{p}{q}$ represent eventually periodic sequences.

An eventually periodic sequence $\{a_n\}$ satisfies that there exists positive integer constants $p, N$ such that for all $k>N$, $$a_k=a_{k+p}$$ In English, this basically means that after a finite number of elements of the sequence which do not repeat, the rest of the elements of the sequence do become periodic.

For example, the sequence $$\{a_n\}=1, 6, 6, 6, \ldots$$ is eventually periodic because for all $k>1$, $$a_k=a_{k+1}$$ I will leave the proof that all rational numbers represent eventually periodic sequences to the reader.

(Video) Periodic Sequences

by Anusha Rahman

In this video, Anusha Rahman goes over how to determine whether a string of numbers is a periodic sequence, how to find the period, and what a real world application of periodic sequences is.

Summary

Periodic sequences: numeric sequences with a repeating pattern

Examples:

$$2, 4, 6, 2, 4, 6, 2, 4, 6$$ $$5, 4, 5, 4, 5, 4...$$ $$\frac{6}{9} = 0.666666$$ $$\frac{3}{7} = 0.428571428571$$

For each of the above examples, what is the repeating pattern?

$$2, 4, 6, 2, 4, 6, 2, 4, 6$$

Repeating pattern: $2, 4, 6$ $$5, 4, 5, 4, 5, 4...$$ Repeating pattern: $5, 4$ $$\frac{6}{9} = 0.666666$$ Repeating pattern: $6$ $$\frac{3}{7} = 0.428571428571$$ Repeating pattern: $4, 2, 8, 5, 7, 1$

We may be asked to find the period. A period is the number of items in a repeated part of a sequence.

Let's go back to the previous examples and find the periods.

$$2, 4, 6, 2, 4, 6, 2, 4, 6$$

• Repeating pattern: $2, 4, 6$
• Period: $3$ $$5, 4, 5, 4, 5, 4...$$
• Repeating pattern: $5, 4$
• Period: $2$ $$\frac{6}{9} = 0.666666$$
• Repeating pattern: $6$
• Period: $1$ $$\frac{3}{7} = 0.428571428571$$
• Repeating pattern: $4, 2, 8, 5, 7, 1$
• Period: $6$

Try the following example problems with the video.

Are the following practice problems periodic sequences? If so, what's the period?

$$5, 7, 8, 9, 10, 11$$ $$\frac{1}{4}$$ $$0.5757575757...$$ $$0.397439743974...$$