Arithmetic Sequences

Introduction: spotting patterns in numbers 📈

students, imagine you are saving money every week, and the amount you save increases by the same number each time. Or imagine rows of seats in a theater, where each new row has exactly the same number more seats than the one before it. Patterns like these are everywhere in real life, and one of the simplest ways to describe them is with an arithmetic sequence.

In IB Mathematics: Applications and Interpretation HL, arithmetic sequences are important because they connect number systems, algebraic representation, and financial modelling. They also build the foundation for more advanced sequence work, such as recursive definitions, series, and technology-supported analysis.

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

explain the key vocabulary used with arithmetic sequences,
find terms in an arithmetic sequence using formulas,
solve problems using the common difference and the $n$th term,
connect arithmetic sequences to financial and real-world contexts,
see how arithmetic sequences fit into the wider topic of Number and Algebra.

An arithmetic sequence is one of the best examples of algebra turning a pattern into a rule. That is a big idea in IB mathematics: moving from repeated calculation to general reasoning. 🧠

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant value is called the common difference, usually written as $d$.

For example, the sequence

$$3, 7, 11, 15, 19, \dots$$

is arithmetic because each term increases by $4$. Here, the common difference is $d=4$.

We use these terms:

$a_1$ = first term,
$a_2$ = second term,
$a_n$ = the $n$th term,
$d$ = common difference.

If a sequence decreases by the same amount each time, it is still arithmetic. For example,

$$20, 17, 14, 11, 8, \dots$$

has common difference $d=-3$.

A good way to think about an arithmetic sequence is “add the same number each time.” This is different from a geometric sequence, where you multiply by the same number each time. Knowing the difference between these two ideas is very important in Number and Algebra.

Example 1: identifying an arithmetic sequence

Consider the sequence

$$5, 9, 13, 17, 21, \dots$$

Check the differences:

$9-5=4$
$13-9=4$
$17-13=4$
$21-17=4$

Since the difference is always $4$, the sequence is arithmetic with $a_1=5$ and $d=4$.

Now compare that with

$$2, 6, 12, 20, 30, \dots$$

The differences are $4, 6, 8, 10, which are not constant. So this is not arithmetic.

The $n$th term formula

One of the most useful results for arithmetic sequences is the formula for the $n$th term:

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

This formula lets you find any term without writing out all the terms before it.

Why does it work? Starting from the first term $a_1$, each new term adds $d$. To get to the $n$th term, you add $d$ a total of $n-1$ times. That is why the formula uses $(n-1)d$.

Example 2: finding a term far along the sequence

Suppose an arithmetic sequence begins with $a_1=8$ and has common difference $d=3$.

Find $a_{10}$.

Use the formula:

$$a_{10}=8+(10-1)3$$

$$a_{10}=8+27=35$$

So the 10th term is $35$.

This is powerful because you did not need to list all ten terms one by one. In IB Mathematics, efficiency matters, especially when modelling or checking results using technology.

Example 3: finding the common difference from a formula

Suppose

$$a_n=12+5(n-1)$$

This is already written in arithmetic sequence form. From the formula, $a_1=12$ and $d=5$.

If a formula is given in a different algebraic form, you may need to expand it first.

For example,

$$a_n=4n-1$$

Rewrite it as:

$$a_n=4n-1=4(n-1)+3$$

So the sequence has first term $a_1=3$ and common difference $d=4$.

This shows how algebraic manipulation helps you recognize patterns. That skill is central to this topic. ✨

Recursive and explicit thinking

Arithmetic sequences can be described in two main ways: explicitly and recursively.

An explicit rule gives a direct formula for the $n$th term. For arithmetic sequences, this is usually

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

A recursive rule defines each term using the previous term. It may look like:

$$a_n=a_{n-1}+d,\quad n\ge 2$$

with a starting value such as $a_1=7$.

Example 4: recursive description

If $a_1=10$ and $d=-2$, then the sequence can be written recursively as:

$$a_1=10,\quad a_n=a_{n-1}-2,\quad n\ge 2$$

The first few terms are:

$$10, 8, 6, 4, 2, \dots$$

Recursive definitions are useful when a sequence is built step by step, such as a monthly allowance decreasing by the same amount after each deduction. Explicit formulas are useful when you want a specific term quickly.

IB questions may ask you to move between these two forms. That is why it is helpful to understand both, not just memorize one formula.

Arithmetic sequences in real-world modelling 💰

Arithmetic sequences often appear in financial and practical situations when change is linear, meaning the amount changes by a constant difference.

Example 5: saving money each week

students saves $15$ in the first week and then increases the amount saved by $5$ each week.

This gives the sequence:

$$15, 20, 25, 30, 35, \dots$$

Here, $a_1=15$ and $d=5$.

Find the amount saved in week $12$.

$$a_{12}=15+(12-1)5$$

$$a_{12}=15+55=70$$

So students saves $70$ in week $12$.

This kind of model is realistic when someone gradually increases savings or spends more in a steady pattern. However, it is important to remember that real-life data is not always perfectly arithmetic. A model is useful when it captures the main pattern well enough.

Example 6: seating in a hall

A school auditorium has $18$ seats in the first row, and each new row has $3$ more seats than the row before it.

Then the number of seats per row is:

$$18, 21, 24, 27, \dots$$

The $n$th row has

$$a_n=18+(n-1)3$$

If there are $20$ rows, the last row has

$$a_{20}=18+19\cdot 3=75$$

So the 20th row has $75$ seats.

This type of problem shows why arithmetic sequences matter in planning and design. 📚

Working with sums of arithmetic sequences

Sometimes IB questions go beyond individual terms and ask for the total of several terms. The sum of the first $n$ terms is written as $S_n$.

The formula is:

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

or equivalently

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

This formula is useful when you need a total amount, such as total savings, total seating, or total cost.

Example 7: total savings

Using the saving pattern from earlier, find the total saved in the first $12$ weeks.

We have $a_1=15$, $d=5$, and $a_{12}=70$.

$$S_{12}=\frac{12}{2}(15+70)$$

$$S_{12}=6\cdot 85=510$$

The total saved is $510$.

This shows a key IB skill: using sequence formulas to model cumulative change.

Common mistakes and how to avoid them

Arithmetic sequence questions can be simple once the pattern is clear, but there are some common errors:

forgetting that the common difference can be negative,
confusing $a_n$ with $S_n$,
using $n$ instead of $n-1$ in the $n$th term formula,
assuming a pattern is arithmetic without checking the differences,
mixing up arithmetic and geometric sequences.

A strong habit is to write out the first few differences before deciding. If the difference is constant, the sequence is arithmetic. If not, it is something else.

Technology can help too. A graphing calculator, spreadsheet, or CAS tool can list terms, check patterns, and compare models. But you should still understand the reasoning behind the formulas. IB Mathematics values both calculation and interpretation.

Conclusion

Arithmetic sequences are a core idea in Number and Algebra because they show how repeated addition can be described with general algebraic rules. students, you have seen that an arithmetic sequence has a constant common difference, that the $n$th term is given by

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

and that the sum of the first $n$ terms is

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

These ideas are useful in practical situations such as saving money, counting seats, and modelling steady change. They also strengthen algebraic thinking by connecting patterns, formulas, and real-world interpretation. Understanding arithmetic sequences prepares you for more advanced sequence and series work in IB Mathematics: Applications and Interpretation HL.

Study Notes

An arithmetic sequence has a constant difference between consecutive terms.
The constant difference is called the common difference and is written as $d$.
The first term is written as $a_1$, and the $n$th term is written as $a_n$.
The explicit formula is $a_n=a_1+(n-1)d$.
The recursive form is $a_n=a_{n-1}+d$, for $n\ge 2$.
The sum of the first $n$ terms is $S_n=\frac{n}{2}(a_1+a_n)$.
Arithmetic sequences model situations with steady additive change.
Always check the differences before deciding that a sequence is arithmetic.
A negative common difference means the sequence decreases.
Arithmetic sequences are part of the broader IB topic of Number and Algebra because they combine numerical patterns, algebraic formulas, and modelling.