Arithmetic sequences and series

Q: What is an arithmetic sequence and how is it defined?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms, called the common difference ($d$), is constant.

Q: What is an arithmetic series?

An arithmetic series is the sum of the terms of an arithmetic sequence, denoted by $S_n$.

Q: How do you interpret sigma notation ($\sum$) for a series?

The bottom number is the starting index, the top number is the ending index, and the expression is the rule used to generate each term to be added.

Q: What is the difference between $n$ and $u_n$ in exam questions?

$n$ is the position (e.g., "which term?"), while $u_n$ is the value of the term at that position (e.g., "what is the value?").

Q: Why is the sign of the common difference ($d$) critical in calculations?

If a sequence is decreasing, $d$ must be negative ; using a positive value will result in incorrect term values and sums.

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: Sep 7, 2026

Welcome to Arithmetic Sequences and Series

Ever noticed a pattern that just keeps adding up? Whether it’s saving $\$10$ more every week, the increasing distance of a staircase, or even the way seats are arranged in a stadium, you are looking at Arithmetic Sequences. In this chapter, we will explore how to identify these patterns, calculate any term in a list, and find the total sum of these numbers quickly.

Don't worry if math patterns feel like a puzzle right now. We’re going to break them down into simple steps that work every single time!

1. What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. We call this constant value the common difference, denoted by the letter $d$.

Key Terms to Know:

$u_1$: The first term in the sequence.
$d$: The common difference (what you add to get to the next term).
$n$: The position of the term (e.g., $n=5$ means the 5th term).
$u_n$: The value of the $n^{th}$ term.

Example: Look at the sequence $5, 8, 11, 14, ...$
Here, the first term $u_1 = 5$.
To get from 5 to 8, we add 3. To get from 8 to 11, we add 3.
So, the common difference $d = 3$.

Quick Tip: If the numbers are going down (e.g., $10, 7, 4, ...$), the common difference is negative. In this case, $d = -3$.

Key Takeaway: If you can find the first term and the common difference, you have the "DNA" of the entire sequence!

2. Finding Any Term: The $n^{th}$ Term Formula

What if you wanted to find the $100^{th}$ term? You wouldn't want to keep adding $d$ a hundred times! Instead, we use this formula from your IB Formula Booklet:

$u_n = u_1 + (n - 1)d$

Why $(n-1)$?
Think about it: To get to the 2nd term, you add the difference once. To get to the 3rd term, you add it twice. To get to the $n^{th}$ term, you always add the difference one less time than the position number.

Step-by-Step Example:
Find the $20^{th}$ term of the sequence $4, 10, 16, 22, ...$
1. Identify $u_1 = 4$.
2. Identify $d = 10 - 4 = 6$.
3. We want the $20^{th}$ term, so $n = 20$.
4. Plug into the formula: $u_{20} = 4 + (20 - 1) \times 6$
5. Calculate: $u_{20} = 4 + (19 \times 6) = 4 + 114 = 118$.

3. Arithmetic Series: Adding It All Up

A series is what you get when you add the terms of a sequence together. The sum of the first $n$ terms is denoted by $S_n$.

There are two formulas for the sum in your booklet. Use the one that fits the information you have:

Formula A (Use if you know the first and last terms):
$S_n = \frac{n}{2}(u_1 + u_n)$

Formula B (Use if you know the first term and the difference):
$S_n = \frac{n}{2}(2u_1 + (n - 1)d)$

Analogy: Imagine a pile of logs. The top row has 1 log, the next has 2, the next has 3. To find the total logs ($S_n$), you just need to know how many rows there are ($n$) and how many logs are in the first and last rows!

Key Takeaway: Always check your formula booklet during the exam to ensure you don't swap a sign or miss a bracket!

4. Sigma Notation ($\sum$)

Sometimes, the IB will use a shorthand symbol called Sigma ($\sum$) to ask you to find a sum. It looks intimidating, but it's just a set of instructions.

$\sum_{r=1}^{k} (\text{expression})$

The bottom number ($r=1$) tells you where to start.
The top number ($k$) tells you where to stop.
The expression tells you the rule for the sequence.

Example: $\sum_{r=1}^{5} (2r + 1)$ means:
When $r=1$: $2(1)+1 = 3$
When $r=2$: $2(2)+1 = 5$
When $r=3$: $2(3)+1 = 7$
...and so on, until $r=5$. Then add those results together: $3 + 5 + 7 + 9 + 11 = 35$.

5. Applications and Modeling

In the real world, things aren't always "perfectly" arithmetic. The IB syllabus requires you to interpret models that might be approximately arithmetic.

Real-World Scenarios:

Linear Depreciation: An object losing the same amount of value each year.
Simple Interest: Earning interest only on the original amount (though Compound Interest is a different chapter!).
Physical patterns: Distances traveled in training (e.g., running $500m$ further each day).

Imperfect Models: If a question gives you data points that don't have an exact common difference, you might be asked to find an "average" difference or identify if an arithmetic model is appropriate. If the differences are very close (e.g., $4.1, 3.9, 4.0$), an arithmetic model is often a good fit.

Did you know? The famous mathematician Gauss supposedly discovered the sum formula as a child when his teacher asked the class to add all numbers from 1 to 100 to keep them busy. He realized he could pair them up ($1+100=101, 2+99=101$) and finished in seconds!

6. Common Mistakes to Avoid

1. Confusing $n$ and $u_n$: Remember that $n$ is the "seat number" and $u_n$ is the "person sitting in it." If a question asks for the value of the 10th term, they want $u_{10}$. If they ask which term has a value of 50, they want you to solve for $n$.

2. The $(n-1)$ Trap: When calculating $u_n$, many students forget to subtract 1 from $n$ before multiplying by $d$. Always double-check your brackets!

3. Negative Differences: If the sequence is decreasing, $d$ must be negative. If you use a positive $d$ for a decreasing sequence, your sum and terms will be way off.

4. Calculator vs. No Calculator:
- In Paper 1, you must be comfortable with basic arithmetic and algebraic rearrangement.
- In Paper 2, you can use your GDC (Graphic Display Calculator) to solve equations or find sums quickly. Practice using the "Sequence" or "Solver" functions on your device.

Quick Review:
- Arithmetic: Constant addition/subtraction ($d$).
- $n^{th}$ Term: $u_n = u_1 + (n-1)d$.
- Sum: $S_n = \frac{n}{2}(\text{first} + \text{last})$.
- Sigma: Just a fancy way to say "add these up."

Next Chapter Suggestion: If you've mastered adding the same amount, head over to Geometric Sequences to learn about patterns that multiply!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is an arithmetic sequence and how is it defined?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms, called the common difference ($d$), is constant.

What do the symbols $u_1$, $d$, $n$, and $u_n$ represent in the context of sequences?

$u_1$ is the first term, $d$ is the common difference, $n$ is the position number of the term, and $u_n$ is the value of the term at that position.

What is the formula for the $n^{th}$ term of an arithmetic sequence?

The formula is $u_n = u_1 + (n - 1)d$.

What is an arithmetic series?

An arithmetic series is the sum of the terms of an arithmetic sequence, denoted by $S_n$.

Which two formulas can be used to calculate the sum of the first $n$ terms ($S_n$)?

$S_n = \frac{n}{2}(u_1 + u_n)$ (if the last term is known) or $S_n = \frac{n}{2}(2u_1 + (n - 1)d)$ (if only the difference is known).

How do you interpret sigma notation ($\sum$) for a series?

The bottom number is the starting index, the top number is the ending index, and the expression is the rule used to generate each term to be added.

What is linear depreciation in the context of arithmetic modeling?

Linear depreciation occurs when an object loses a fixed, constant amount of value over each time period, which can be modeled by an arithmetic sequence with a negative $d$.

What is the difference between $n$ and $u_n$ in exam questions?

$n$ is the position (e.g., "which term?"), while $u_n$ is the value of the term at that position (e.g., "what is the value?").

Why is the sign of the common difference ($d$) critical in calculations?

If a sequence is decreasing, $d$ must be negative; using a positive value will result in incorrect term values and sums.

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