Arithmetic sequences and series

Q: What real-world financial concept is modeled by an arithmetic sequence?

Simple Interest is modeled by an arithmetic sequence because the balance increases by a fixed amount (a constant difference) every period.

Mathematics - Applications and Interpretation · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Arithmetic Sequences and Series

Welcome to the world of Arithmetic Sequences! If you have ever counted by fives ($5, 10, 15, 20...$) or watched a taxi meter go up by the same amount every kilometer, you have already used arithmetic sequences. In this chapter, we will learn how to describe these patterns mathematically, find missing numbers in a long list, and add them all up quickly.

This topic is part of Topic 1: Number and Algebra. While we are focusing on "steady" patterns here, you will later see how these differ from Geometric Sequences (where we multiply instead of add) in the next chapter.

1. What is an Arithmetic Sequence?

An Arithmetic Sequence is simply a list of numbers where the difference between one term and the next is always the same constant value. We call this constant value the common difference, denoted by the letter $d$.

To find the common difference $d$, you just subtract any term from the one after it:
$d = u_{n+1} - u_n$

Example: Consider the sequence $3, 7, 11, 15, ...$
Here, the first term $u_1 = 3$.
The common difference $d = 7 - 3 = 4$.
Since we keep adding $4$ every time, this is an arithmetic sequence.

Quick Tip: If the numbers are getting smaller (e.g., $10, 7, 4, 1...$), the common difference $d$ will be a negative number (in this case, $d = -3$).

Key Takeaway:

An arithmetic sequence "grows" or "shrinks" by adding the same amount every time.

2. Finding the $n$th Term

What if you wanted to find the $100$th number in a sequence? You wouldn't want to add $d$ a hundred times! Instead, we use the general term formula:

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

Breaking down the formula:

$u_n$: The value of the term you are looking for (the "$n$th" term).
$u_1$: The very first number in the list.
$n$: The position of the term (e.g., for the $10$th term, $n = 10$).
$d$: The common difference.

Why $(n-1)$? Think of it this way: to get to the 2nd term, you add the difference once. To get to the 3rd term, you add it twice. You always add the difference one fewer time than the position you are trying to reach!

Example: Find the $20$th term of the sequence $5, 8, 11, ...$
1. Identify $u_1 = 5$.
2. Identify $d = 8 - 5 = 3$.
3. We want the $20$th term, so $n = 20$.
4. Plug into the formula: $u_{20} = 5 + (20 - 1) \times 3$
5. Calculate: $u_{20} = 5 + (19 \times 3) = 5 + 57 = 62$.

3. Arithmetic Series (Adding them up)

When we add the terms of a sequence together, it becomes a series. The sum of the first $n$ terms is denoted by $S_n$. There are two formulas you can use, depending on what information you have:

Formula 1 (If you know the first and last term):
$S_n = \frac{n}{2}(u_1 + u_n)$

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

Analogy: Imagine you have a staircase. The first formula is like taking the height of the bottom step ($u_1$) and the top step ($u_n$), averaging them, and multiplying by the number of steps. It’s a shortcut to adding every single step height!

Key Takeaway:

Use Formula 1 if you have the "last" term. Use Formula 2 if you only have the "difference."

4. Sigma ($\Sigma$) Notation

In your IB exams, you will see a funny-looking Greek letter: $\Sigma$. This is the Sigma symbol, and it just means "Sum everything up."

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

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

Example: $\sum_{r=1}^{4} (2r + 1)$ means:
When $r=1$: $2(1)+1 = 3$
When $r=2$: $2(2)+1 = 5$
When $r=3$: $2(3)+1 = 7$
When $r=4$: $2(4)+1 = 9$
Total Sum: $3 + 5 + 7 + 9 = 24$.

Technology Tip: Your Graphic Display Calculator (GDC) has a $\Sigma$ function! Learn to find it in the math menu—it will save you a lot of time on Paper 1 and Paper 2.

5. Real-Life Applications and Modelling

Arithmetic sequences are great for modeling "linear" growth. Some examples include:

Simple Interest: If you earn $\$10$ interest every year, your balance follows an arithmetic sequence.
Salaries: A job that pays $\$40,000$ with a fixed $\$2,000$ raise every year.
Physical Distance: An object moving at a constant speed.

Predicting with "Imperfect" Models

In the real world, data isn't always a perfect arithmetic sequence. For example, a plant might grow about $2$ cm every week, but one week it's $2.1$ cm and the next it's $1.9$ cm. In these cases, we use the average difference as our $d$ to make a "best fit" prediction for the future.

Don't worry if the data looks messy! If the question asks for a "linear model" or an "arithmetic model," look for the average amount the values are increasing by.

6. Common Mistakes to Avoid

Confusing $n$ and $u_n$: Remember that $n$ is the position (e.g., Day 5) and $u_n$ is the actual value (e.g., $\$50$).
The Sign of $d$: If the sequence is decreasing, $d$ must be negative. If you forget the minus sign, your sum will be huge instead of small!
The $(n-1)$ slip-up: Always remember to subtract $1$ from the position before multiplying by $d$.
Calculator Notation: While your GDC might show $5.2E30$, never write this on your exam paper. Always convert it to standard form: $5.2 \times 10^{30}$.

Quick Review Box:

Arithmetic $n$th term: $u_n = u_1 + (n - 1)d$
Common difference: $d = u_2 - u_1$
Sum of terms: $S_n = \frac{n}{2}(\text{first} + \text{last})$
Key concept: Constant addition or subtraction.

Quick check

Can you answer these now?

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

What is the definition of an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same constant value, known as the common difference, $d$.

How do you calculate the common difference $d$ of an arithmetic sequence?

Subtract any term from the term that follows it using the formula: $d = u_{n+1} - u_n$.

What is the formula for the general term ($n$th term) of an arithmetic sequence?

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

When should you use the arithmetic series formula $S_n = \frac{n}{2}(u_1 + u_n)$?

Use this formula when you know the number of terms ($n$), the first term ($u_1$), and the final term ($u_n$) being summed.

What is the arithmetic series formula used when the last term is unknown?

$S_n = \frac{n}{2}(2u_1 + (n - 1)d)$

In Sigma ($\Sigma$) notation, what do the top and bottom numbers represent?

The bottom number is the starting index (where you begin), and the top number is the stopping index (where you end).

What real-world financial concept is modeled by an arithmetic sequence?

Simple Interest is modeled by an arithmetic sequence because the balance increases by a fixed amount (a constant difference) every period.

What is the difference between $n$ and $u_n$ in the context of sequences?

$n$ refers to the position of a term (the index), while $u_n$ refers to the actual value of the term at that position.

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Arithmetic sequences and series

Introduction to Arithmetic Sequences and Series

1. What is an Arithmetic Sequence?

Key Takeaway:

2. Finding the \(n\)th Term

3. Arithmetic Series (Adding them up)

Key Takeaway:

4. Sigma (\(\Sigma\)) Notation

5. Real-Life Applications and Modelling

Predicting with "Imperfect" Models

6. Common Mistakes to Avoid

Quick Review Box:

Can you answer these now?

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