What is a Sequence - Definition & Examples

(Video) Intro to Series and Sequences

by Eddie Woo

Here is a fantastic video by Eddie Woo where he goes over the basics of sequences and series.

In this video he goes over:

$$\\\begin{align} &\circ\text{ Sets}\\ &\circ\text{ Sequences}\\ &\circ\text{ Series}\\ &\circ\text{ Terms}\\ &\circ\text{ How to Define a Sequence or Series}\\ &\quad\cdot\text{ Show the Pattern}\\ &\quad\cdot\text{ Algebraic Expression}\\ &\quad\cdot\text{ Recursive Definition}\\ &\quad\quad-\text{ Fibonacci Sequence} \end{align}$$

In the rest of the explanation, we will go over sets, sequences, series, and terms.

Sets

A set is just a collection of numbers (where there are no duplicates). We indicate it with curly braces. $$\begin{align} &\text{Set of Whole Numbers: }\{1,2,3,\cdot\cdot\cdot\}\\ &\text{Set of Odd Numbers Less Than 10: }\{3,5,1,7,9\} \end{align}$$ Sets can infinite which is denoted by the dot dot dot, $\cdot\cdot\cdot$, as in the the set of whole numbers. Sets can also be finite or limited, as in the set of odd numbers less than ten.

It is important to note that these two sets are considered equal: $$\{3,5,1,7,9\}=\{1,3,5,7,9\}$$ This is because they contain the same numbers. Order does not matter with sets.

Sequences

A sequence adds order into the picture. You can call it an ordered set. Also, we drop the curly braces, and duplicates are allowed. Now, these two sequences are different: $$\begin{align} &\text{Descending: }9,7,5,3,1\\ &\text{Ascending: }1,3,5,7,9 \end{align}$$

(Note: You can also write sequences inside parentheses!)

Series

A series is the sum of an infinite sequence. In other words, a series is just the addition of all the terms in an infinite sequence.

For example, the following series corresponds to the sequence of odd numbers starting at $1:$ $$1+3+5+7+9+11+\cdots$$

You can also find partial sums of sequences. This is kind of like finding the sum of a series, but you only find the sum of the first $n$ terms.

Terms

One way to describe sequences or series is to talk about their terms. A term is just each individual number. We denote each term with the letter $T$ and a subscript number. $$\begin{align} \text{Term }1:&\quad T_1\\ \text{Term }2:&\quad T_2\\ n^{\text{th}}\text{ term}:&\quad T_n\\ \end{align}$$

(Video) Finding Patterns

by Textbook Tactics

This video by Textbook Tactics takes a look at mathematical patterns called sequences.

Summary

The first example is looking at the distance traveled going $50 \text{ mph}.$ We start by listing the first hour and the total distanced traveled. $$\begin{array}{c|c} \text{Hour}&1\\ \hline \text{Distance}&50 \end{array}$$ After another hour, we've driven another $50 \text{ miles}.$ So for $2$ hours the total distance is $100\text{ miles}.$ $$\begin{array}{c|cc} \text{Hour}&1&2\\ \hline \text{Distance}&50&100 \end{array}$$ Let's continue this until we reach $5$ hours. $$\begin{array}{c|cc} \text{Hour}&1&2&3&4&5\\ \hline \text{Distance}&50&100&150&200&250 \end{array}$$ Notice, there's a pattern between these distances. To go from left to right, we add $50$ each time.

Because these total distances have a pattern (a change of $50$) and they're written in an order, we can call these numbers a sequence.

$$\begin{align} \text{Sequence}&: (50, 100, 150, 200, 250)\\ &\quad a_1\quad a_2\quad a_3\quad a_4\quad a_5 \end{align}$$

The small subscript for each term (i.e. the $1$ in $a_1$) tells us the place in the sequence that the term belongs.

We can use this general form of a sequence to refer to the sequence above. The first term, $a_1$, will be $50$. The second term, $a_2$, will be $100$. And so on.

We use $n$ to refer to some general term in the sequence. We call this the $n$-th term, $a_n$.

Example

Let's next look at the sequence: $$3\quad 9\quad 27\quad 81\quad 243$$ First, we want to find the pattern. We want to ask how can we get from the first term, $3$, to the second term, $9$. Two quick ways would be by adding $6$ or multiplying by $3$.

Next, look at how we get from the second term, $9$, to the third term, $27$. If we were to add $6$ to the second term, this wouldn't get us to the third term. Therefore, we can tell this isn't the pattern. If we multiply $3$ to the second term, this would bring us to the third term. Therefore, the pattern here is multiplying each term by $3$.

So, we found the second term by taking the first term and multiplying it by $3.$ $$a_2=a_1\times 3$$ And similarly, we found the third term by taking the second term and multiplying it by $3.$ $$a_3=a_2\times3$$ Therefore, we could find some $n$-th term by taking the term before it and multiplying it by $3.$ This can be written as: $$\boxed{a_n=a_{n-1}\times 3}$$ This is a recursive formula that describes the sequence.

There are two ways we can define a sequence.

The first is using a recursive formula. This is a sequence where each term is dependent on the previous terms. In the example above, we used a recursive formula since each term depended on the term before it.

The other is a explicit formula. This is a sequence where each term is a function of its position (the little subscript $n$ in $a_n$). We could write the above sequence explicitly with: $$a_n=3^n$$ The first term is defined by $3^1$, the second is defined by $3^2$, and so on.

Sequences

Check out this graphic! It covers the main characteristics of sequences.

What is a Sequence?

A sequence is a list of objects (often numbers) where the order of the objects matters and repetition is allowed. Ordering and repetition are what separates a sequence from a set. There are many types of sequences, and we'll go over some in this explanation.

Finite or Infinite

Sequences can either be finite or infinite. Here are some examples: $$\begin{align} \text{Finite Sequence}&: (P, I, T, T, S, B, U, R, G, H)\\ \text{Infinite Sequence}&: (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...)\\ \end{align}$$

(In the finite sequence, notice that the elements are letters and that there is repetition. In both sequences, order matters. If you were to rewrite the finite sequence as $(H, G, R, U, B, S, T, T, I, P),$ it would be a completely different sequence. If you were to rewrite the infinite sequence as $(10, 9, 8, 7, 6, 5, 4, 3, 2, 1, ...),$ it would be a completely different sequence.)

How Do We Denote Sequences?

There are three main ways to specify a sequence and its elements.

Listing: One way to denote a sequence is to list its elements between parentheses (or curly braces)! A couple good examples of listing are the finite and infinite sequences from earlier in the explanation, $(P, I, T, T, S, B, U, R, G, H)$ and $(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...).$ If you are listing an infinite sequence, start with first few terms that people show the pattern, then add some ellipses at the end to denote an infinite sequence!

Indexing: Sometimes we write each element of a sequence as a function of its position. This helps us denote a sequence in a more concise way. Before we get into this method, let's briefly discuss positions of elements in sequences using an ordered list of numbers.

Say we have the sequence of even positive integers: $$(2, 4, 6, 8, 10, ...)$$

We can use the notation $a_n$ to denote the $n^{\text{th}}$ element of the sequence. Since $2$ is the $1^{\text{st}}$ element, $a_1=2.$ Since $4$ is the $2^{\text{nd}}$ element, $a_2=4.$ Since $6$ is the $3^{\text{rd}}$ element, $a_3=6.$ And so on. Since the $n^{\text{th}}$ element of our sequence is $a_n,$ and since we use the natural numbers to count our elements, we can denote our sequence in the following general way: $$\text{Sequence} = (a_n)_{n\in \mathbb{N}}$$

(Note, you don't have to use the natural numbers to count the positions of elements in your sequence from $1$ onwards. Another common approach is to start at $0.$)

However, this is not too helpful, because it doesn't tell us the pattern the sequence follows! Let's figure out how to write each element of a sequence as a function of its position. Looking at our sequence of even positive integers, $(2, 4, 6, 8, 10, ...),$ we notice that:

$$\begin{align} a_1 &= 2 = 2\cdot 1\\ a_2 &= 4 = 2\cdot 2\\ a_3 &= 6 = 2\cdot 3\\ a_4 &= 8 = 2\cdot 4\\ a_5 &= 10 = 2\cdot 5\\ &\cdots\\ a_n &=2n= 2\cdot n \end{align}$$

In other words, the $n^{\text{th}}$ element of this sequence is $2n.$ This pattern can help us denote the sequence of even positive integers in a very concise way: $$(2n)_{n\in \mathbb{N}}$$

Each term of the sequence can be found using the explicit formula: $$a_n = 2n$$

Here are some other examples: $$\begin{align} \text{Sequence of Odd Positive Integers} &: (1, 3, 5, 7, 9, ...) = (2n-1)_{n\in \mathbb{N}}\\ \text{Sequence of Natural Numbers} &: (1, 2, 3, 4, 5, ...) = (n)_{n\in \mathbb{N}}\\ \text{Sequence of Squares of Natural Numbers} &: (1, 4, 9, 16, 25, ...) = (n^2)_{n\in \mathbb{N}}\\ \end{align}$$

Recursion: We won't go too much in depth here about recursion and recursive formulas, but check out the next topic to get more information! With recursive sequences, you define at least one initial element of the sequence, and then you define a recurrence relation. A recurrence relation is a formula that relates each element to previous elements.

A good example of a recurrence relation is the one that defines the Fibonacci sequence!

$$\begin{align} \text{Fibonacci Sequence} &: (0, 1, 1, 2, 3, 5, 8, 13, ...)\\ \text{Recurrence Relation} &: F_n = F_{n-1}+F_{n-2}, \text{ with } F_0=0, F_1=1 \end{align}$$