Arithmetic Sequences and Series

Introduction

students, imagine you are saving money each week 📈. In week 1 you save $5$, in week 2 you save $8$, in week 3 you save $11$, and so on. The amount is increasing by the same number each time. This is the main idea behind an arithmetic sequence: a list of numbers where the difference between consecutive terms is constant.

In this lesson, you will learn how to recognize arithmetic sequences, find important terms such as the first term and common difference, and calculate the sum of many terms in an arithmetic series. These ideas are useful in real life for saving plans, salary increases, stair-step patterns, and any situation where change happens by a fixed amount 🚀.

Learning objectives

By the end of this lesson, students, you should be able to:

explain the meaning of arithmetic sequence and arithmetic series;
identify the first term, common difference, and general term;
use formulas to find terms and sums;
solve IB-style problems involving patterns and real-world situations;
connect this topic to algebraic reasoning and number patterns.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between one term and the next is always the same. That constant difference is called the common difference, written as $d$.

For example, the sequence $4, 7, 10, 13, 16, \dots$ is arithmetic because each term increases by $3$. Here:

first term $a_1 = 4$,
common difference $d = 3$.

The next terms are found by adding $d$ each time. So if you know one term, you can build the rest of the sequence.

Arithmetic sequences can also go down. For example, $20, 17, 14, 11, \dots$ has common difference $d = -3$.

A key idea in IB Mathematics Analysis and Approaches SL is that you should be able to describe a pattern algebraically, not just list the numbers. That means using symbols such as $a_n$ for the $n$th term.

The general term

The $n$th term of an arithmetic sequence is given by

$$a_n = a_1 + (n-1)d$$

where:

$a_n$ is the $n$th term,
$a_1$ is the first term,
$d$ is the common difference,
$n$ is the term number.

This formula is important because it lets students find any term directly without writing out all earlier terms.

Example 1

Find the $10$th term of the arithmetic sequence $3, 7, 11, 15, \dots$

Here, $a_1 = 3$ and $d = 4$.

Using the formula:

$$a_{10} = 3 + (10-1)4 = 3 + 36 = 39$$

So the $10$th term is $39$.

Example 2

A sequence has $a_1 = 12$ and $d = -2$. Find $a_6$.

Using

$$a_n = a_1 + (n-1)d$$

we get

$$a_6 = 12 + (6-1)(-2) = 12 - 10 = 2$$

So the $6$th term is $2$.

Arithmetic series: adding the terms

A series is the sum of the terms of a sequence. When you add the terms of an arithmetic sequence, you get an arithmetic series.

For example, the series

$$4 + 7 + 10 + 13 + 16$$

is the sum of the arithmetic sequence $4, 7, 10, 13, 16.

If there are $n$ terms, the sum is written as $S_n$.

The standard formula for the sum of the first $n$ terms of an arithmetic sequence is

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Another useful form is

$$S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr)$$

Both formulas are correct. Use the one that fits the information you are given.

Why the sum formula works

One clever way to understand the formula is to pair the first and last terms:

first + last,
second + second last,
third + third last.

Each pair has the same total. If there are $n$ terms, then there are $\frac{n}{2}$ such pairs. That is why the formula uses $\frac{n}{2}$.

This is a good example of mathematical reasoning in IB: instead of memorizing blindly, students should also understand the structure behind the formula.

Example 3

Find the sum of the first $8$ terms of $2, 5, 8, 11, \dots$

We know $a_1 = 2$, $d = 3$, and $n = 8$.

First find the $8$th term:

$$a_8 = 2 + (8-1)3 = 23$$

Now use the sum formula:

$$S_8 = \frac{8}{2}(2 + 23) = 4 \cdot 25 = 100$$

So the sum of the first $8$ terms is $100$.

Example 4

Find the sum of the first $15$ terms of the arithmetic sequence with $a_1 = 7$ and $d = 2$.

Use

$$S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr)$$

Substitute the values:

$$S_{15} = \frac{15}{2}\bigl(2(7) + (15-1)2\bigr)$$

$$S_{15} = \frac{15}{2}(14 + 28) = \frac{15}{2}(42) = 315$$

So $S_{15} = 315$.

Solving problems and recognizing patterns

In exams, questions often mix arithmetic sequences with algebra, graphs, or word problems. students should look for a constant difference, then choose the right formula.

A common problem type: finding the number of terms

Suppose a sequence starts with $5$ and increases by $4$ each time. If the last term is $61$, how many terms are there?

We use

$$a_n = a_1 + (n-1)d$$

$$61 = 5 + (n-1)4$$

$$56 = 4(n-1)$$

$$14 = n-1$$

$$n = 15$$

There are $15$ terms.

Real-world example: savings plan 💰

A student saves $10$ in the first week and increases savings by $3$ each week. The weekly savings form the arithmetic sequence $10, 13, 16, 19, \dots$.

If the student wants to know how much is saved in $12$ weeks, first find the $12$th term:

$$a_{12} = 10 + (12-1)3 = 43$$

Then find the total savings:

$$S_{12} = \frac{12}{2}(10 + 43) = 6 \cdot 53 = 318$$

So the total saved in $12$ weeks is $318$.

This kind of question connects arithmetic series to everyday number patterns in the world around us.

Real-world example: seating rows 🎭

A theater has $20$ seats in the first row and each new row has $2$ more seats than the row before it. The numbers of seats per row form an arithmetic sequence. If there are $10$ rows, the total number of seats is

$$S_{10} = \frac{10}{2}\bigl(2(20) + (10-1)2\bigr)$$

$$S_{10} = 5(40 + 18) = 290$$

So the theater has $290$ seats.

Arithmetic sequences in algebra and Number and Algebra

Arithmetic sequences fit naturally into the Number and Algebra topic because they show how numbers change in a predictable way. They also link with symbolic manipulation because you must write formulas, substitute values, and rearrange equations.

You may be asked to:

solve for $n$ using the term formula;
solve for $d$ if some terms are known;
derive expressions for terms or sums;
identify patterns from tables or graphs.

For example, if $a_3 = 11$ and $a_7 = 27$, then

$$a_3 = a_1 + 2d = 11$$

$$a_7 = a_1 + 6d = 27$$

Subtracting gives

$$4d = 16$$

$$d = 4$$

Then

$$a_1 + 2(4) = 11$$

$$a_1 = 3$$

Thus the sequence is $3, 7, 11, 15, \dots$.

This is a strong example of algebraic thinking: using known information to build an unknown pattern.

Common mistakes to avoid

students, these are some frequent errors:

confusing the common difference $d$ with the value of a term;
using $n$ instead of $n-1$ in the term formula;
forgetting that the sum formula needs either $a_n$ or enough information to find it;
mixing arithmetic sequences with geometric sequences, which multiply by a constant ratio instead of adding a constant difference.

A quick check can help: if the sequence changes by adding or subtracting the same number each time, it is arithmetic. If the change is multiplicative, it is not arithmetic.

Conclusion

Arithmetic sequences and series are a key part of Number and Algebra because they show how numerical patterns can be described clearly with symbols and formulas. students, you should now be able to identify arithmetic sequences, find the common difference, write the general term, and calculate sums of terms. These skills are useful in algebra, problem solving, and real-world contexts such as finance and planning.

Understanding the logic behind the formulas is just as important as memorizing them. In IB Mathematics Analysis and Approaches SL, the goal is not only to compute answers but also to explain patterns and justify methods.

Study Notes

An arithmetic sequence has a constant difference between consecutive terms.
The common difference is written as $d$.
The $n$th term formula is $a_n = a_1 + (n-1)d$.
An arithmetic series is the sum of the terms of an arithmetic sequence.
The sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$.
Another sum formula is $S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr)$.
To find the number of terms, use the term formula and solve for $n$.
Arithmetic sequences often appear in savings, seating plans, and other fixed-change situations.
A sequence is arithmetic only if the difference stays constant.
This topic connects algebraic manipulation with number patterns and proof-based reasoning.