Language of Sequences & Series

Hi! Welcome to the world of sequences and series. Don’t be put off by the technical names — at heart, these are just patterns of numbers that follow a rule. Once you learn the simple language used to describe them, every IB question on this topic becomes much easier to read. In this lesson we’ll cover all the words and symbols you need before diving into arithmetic and geometric sequences in the next lessons.

Step 1: What is a sequence?

A sequence is just an ordered list of numbers that follows a rule. The order matters — you can’t shuffle them around.

Here are some examples of sequences:

• 2, 4, 6, 8, 10, …   rule: “start at 2, add 2 each time”
• 1, 4, 9, 16, 25, …   rule: “the square numbers” (1², 2², 3², 4², …)
• 3, 6, 12, 24, 48, …   rule: “start at 3, double each time”
• 10, 7, 4, 1, −2, …   rule: “start at 10, subtract 3 each time”

Each number in the sequence is called a term. So in the sequence 2, 4, 6, 8, 10:

• 2 is the first term
• 4 is the second term
• 6 is the third term
• … and so on

Step 2: The special notation — meet u_n

Mathematicians don’t like writing “the first term”, “the second term”, etc., because it takes too long. So we use a much shorter notation. Each term gets the letter u with a small subscript (the little number underneath) that tells you which term it is.

So:

• u₁ = the 1st term
• u₂ = the 2nd term
• u₃ = the 3rd term
• u_n = the n^th term   (the general term — read it as “u sub n”)

For example, in the sequence 2, 4, 6, 8, 10:

• u₁ = 2
• u₂ = 4
• u₃ = 6
• u₄ = 8
• u₅ = 10

Step 3: The “n^th term formula”

Here’s where things get clever. Instead of writing out the sequence forever, we can give a single formula that lets you find any term you want — without listing all the previous ones.

This is called the n^th term formula, and it’s usually written as u_n = (some expression in n).

For example, suppose the formula is:

To find any term, you just substitute the term number in for n. Watch:

So the sequence is 1, 3, 5, 7, 9, … — the odd numbers! And we can jump straight to the 100th term without writing out the first 99. That’s the power of the formula.

Step 4: What is a series?

Once you understand sequences, a series is super easy. A series is just what you get when you add up the terms of a sequence.

So they’re closely related — a series is what you get when you “add up” a sequence. Whenever a question asks for the “sum of the first n terms”, they want a series.

Step 5: The notation S_n

Just like we used u_n for “the n^th term”, we use S_n as shorthand for “the sum of the first n terms”.

So:

• S₁ = sum of the first 1 term  = u₁
• S₂ = sum of the first 2 terms  = u₁ + u₂
• S₃ = sum of the first 3 terms  = u₁ + u₂ + u₃
• S_n = sum of the first n terms

Let’s try it. For the sequence 1, 3, 5, 7, 9:

Step 6: Quick vocabulary recap

Before the worked examples, let’s lock in all the language we’ve learned. You’ll see these terms in every IB exam question on this topic:

Step 7: Worked examples