Introduction to Sequences and Series

Lists with a Purpose

Mathematics is often about finding patterns. A sequence is one of the simplest ways to organize a pattern. It's just an ordered list of numbers. Each number in the list is called a term.

For example, the list of even numbers is a sequence: 2, 4, 6, 8, 10, ...

The three dots, called an ellipsis, mean the pattern continues forever. This is an infinite sequence. A sequence can also be finite, meaning it has a specific end.

We can label the terms to make them easier to talk about. The first term is often called $a_1$ , the second is $a_2$ , and so on. The 'n-th' term, $a_n$ , is a general formula that tells you what any term in the sequence is. For our even number sequence, the formula is $a_n = 2n$ . If you want the 5th term ( $a_5$ ), you just plug in 5 for $n$ : $2 \times 5 = 10$ .

sequence

noun

An ordered list of numbers, called terms, that follow a specific rule or pattern.

Here are a few more examples of sequences:

Powers of 3: 3, 9, 27, 81, ... (The rule is $a_n = 3^n$ )
Reciprocals: 1, 1/2, 1/3, 1/4, ... (The rule is $a_n = 1/n$ )

From Lists to Sums

Now, what happens if we add up the terms of a sequence? We get a series. A series is simply the sum of the terms in a sequence.

series

noun

The sum of the terms of a sequence.

If we take our sequence of even numbers (2, 4, 6, 8, ...), the corresponding series is 2 + 4 + 6 + 8 + ...

Just like sequences, a series can be finite or infinite. The sum of the first four even numbers is a finite series:

2 + 4 + 6 + 8 = 20

To write series in a compact way, mathematicians use summation notation, which uses the Greek letter sigma, $\Sigma$ .

\sum_{n=1}^{4} 2n

Let's break that down:

The $\Sigma$ means "sum."
$n=1$ at the bottom is the starting point. We begin with the first term.
The 4 at the top is the ending point. We stop at the fourth term.
$2n$ is the formula for the terms we are adding up.

So, this notation is just a shorthand way of writing 2(1) + 2(2) + 2(3) + 2(4), which is 2 + 4 + 6 + 8.

Sequence vs. Series

The distinction is simple but important. A sequence is a list of numbers separated by commas. A series is a sum of numbers separated by plus signs.

Feature	Sequence	Series
What is it?	An ordered list of numbers.	The sum of the numbers in a list.
Notation	Uses commas: $a_1, a_2, a_3, ...$	Uses plus signs: $a_1 + a_2 + a_3 + ...$
Example	1, 4, 9, 16	1 + 4 + 9 + 16
Key Question	What is the next term in the pattern?	What is the total value of all the terms?

Understanding this difference is the first step in exploring the powerful ways we can work with patterns in numbers. Let's check your understanding.

Quiz Questions 1/5

What is the fundamental difference between a sequence and a series?

Quiz Questions 2/5

A sequence is defined by the formula $a_n = 3n + 2$ . What is the 5th term ( $a_5$ )?

These two concepts, sequences and series, are the foundation for many other topics in mathematics, from calculus to financial math.

Sequences and Series Explained