Writing and Recognizing Sequences in Math

Sequences are definitely a fun part of math — we get to play around with lists of numbers and find patterns!

Formally defined, a sequence is an ordered list of numbers, where each number is an individual term of the sequence. For example, $$1, 3, 5, 7, 9, \cdots$$ is a sequence — to be exact, the sequence of odd natural numbers.

In order for us to be able to more easily specify terms within sequences, we can denote the terms using variables and subscripts. For example, we can write out a sequence as follows: $$a_1, a_2, a_3, \cdots , a_{n-1}, a_n, a_{n+1}, \cdots$$ where the nth term of the sequence is written as $a_n$.

If we know the pattern a sequence follows, we can write an explicit formula for the nth term of the sequence.

Additionally, if we are given an explicit formula for a sequence, we can calculate any term of the sequence by plugging the given value of $n$ into the formula. Try it out!

It turns out that we can also write terms of sequences in terms of other terms in the sequence. This results in something called a recursive formula — for example, the following: $$a_n = a_{n-1} + a_{n-2}$$ If we define $a_0 = 0$ and $a_1 = 1$, this results in the famous Fibonacci sequence, which looks like this: $$0, 1, 1, 2, 3, 5, 8, 13, 21, \cdots$$ Recursive forms are nice because they can make life simpler (most of the time). Consider the fact that the explicit formula for the Fibonacci numbers is $$F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\varphi + \varphi^{-1}}$$ where $\varphi = \frac{1+\sqrt{5}}{2}$. To be fair, explicit formulas could look a lot worse, but isn't the recursive form just so much nicer?

To wrap things up, let's try calculating some terms of a recursive sequence!

Now you're ready to check out some different kinds of sequences — why not try out arithmetic sequences, geometric sequences, or periodic sequences?

What is a sequence?

A sequence is a list of things (which are elements or terms of the sequence). The two defining features of a sequence are that:

Order matters. The sequence $1, 2, 3, 4, 5$ is different from the sequence $1, 2, 3, 5, 4$.

Elements can be repeated. The sequence $1, 2, 3, 1, 2$ or even the boring sequence $0, 0, 0, 0, 0$ are both valid sequences.

You can have sequences of anything (for example, a word is a sequence of letters). But we will mostly deal with sequences of numbers.

Formally, you can also say that a sequence is discrete data: a function whose domain is either the set $\{1, 2, \dots, n\}$ (for a sequence of length $n$) or the set of positive integers (for an infinite sequence).

How to specify a sequence

For a short sequence, such as $1, 3, 5, 7, 9$, you can tell someone what it is just by writing down all the terms. If you're lazy, or if the sequence is long, you might look for shortcuts.

If the pattern seems obvious and the sequence is just too long, you could just skip a bunch of terms in the middle and trust that people will understand you. For example, you can probably guess that the sequence $1, 3, 5, \dots, 97, 99$ refers to the sequence of odd numbers less than $100$.

When the pattern isn't obvious, you can give a formula for the terms of the sequence. For example, the same sequence $1, 3, 5, 7, 9$ could be specified by the formula $x_k = 2k-1$, as $k$ goes from $1$ to $5$. (This means that the first term in the sequence is $x_1 = 2\cdot1 - 1 = 1$, and the fourth term is $x_4 = 2\cdot4 - 1 = 7$.) To do this, we had to give the elements of the sequence names, calling them $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$.

Another common trick is to make the formula recursive: defining elements in terms of previous elements. Yet another way to describe the sequence $1, 3, 5, 7, 9$ is to say that $x_1 = 1$ and then $x_k = x_{k-1} + 2$ as $k$ goes from $2$ to $5$. For $k=2$, this tells us that $x_2 = x_1 + 2 = 1 + 2 = 3$. We can then take $k=3$ to see that $x_3 = x_2 + 2 = 3 + 2 = 5$, and so on.

Of course, there are as many ways to specify the sequence $1, 3, 5, 7, 9$ mathematically as there are to explain to someone what the terms of the sequence are. But the methods here are the most common, so it's worth becoming familiar with them.