Arithmetic sequences and series

IB Mathematics: Analysis and Approaches SL· 40 min read

1. Introduction to Arithmetic Sequences★☆☆☆☆⏱ 10 min

📘 Definition

Arithmetic Sequence

$u_{n}$ = nth term, $d$ = common difference

An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, calculated as $d = u_{n + 1} - u_{n}$ for all $n$ .

Example:

2, 5, 8, 11,... is arithmetic with $d = 3$

Arithmetic sequences can be increasing (positive $d$ ), decreasing (negative $d$ ), or constant ( $d = 0$ , all terms equal). Always confirm the difference is constant across the whole sequence to classify it as arithmetic.

📐 Worked Example

Determine if the sequence $18, 14, 10, 5, ...$ is arithmetic.

1
Calculate the difference between the first two terms:
2
$u_2 - u_1 = 14 - 18 = -4$
3
Calculate the difference between the next two terms:
4
$u_3 - u_2 = 10 - 14 = -4$
5
Calculate the next difference to confirm:
6
$u_4 - u_3 = 5 - 10 = -5$
7
Since the difference is not constant across all consecutive terms:
8
The sequence is not arithmetic.

Exam tip:

Always check at least two pairs of consecutive terms to confirm a sequence is arithmetic.

2. The nth Term of an Arithmetic Sequence★★☆☆☆⏱ 15 min

📘 Definition

nth Term Formula

The nth term of any arithmetic sequence is given by the formula below, derived by adding the common difference $(n - 1)$ times to the first term:

Example:

$u_{n} = u_{1} + (n - 1) d$

This formula can be rearranged to find any unknown value ( $u_{1}$ , $d$ , or $n$ ) when the other three are known. This is one of the most commonly tested skills for this sub-topic in IB exams.

📐 Worked Example

The 4th term of an arithmetic sequence is 12, and the 9th term is 27. Find $u_{1}$ and $d$ .

1
Write equations for both terms using the nth term formula:
2
$\begin{align} u_4 &= u_1 + 3d = 12 \\ u_9 &= u_1 + 8d = 27 \end{align}$
3
Subtract the first equation from the second to eliminate $u_{1}$ :
4
$(u_1 + 8d) - (u_1 + 3d) = 27 - 12 \\ 5d = 15 \\ d = 3$
5
Substitute $d = 3$ back into the first equation to find $u_{1}$ :
6
$u_1 + 3(3) = 12 \\ u_1 = 12 - 9 = 3$
7
Final result: $u_{1} = 3$ , $d = 3$

3. Sum of an Arithmetic Series★★☆☆☆⏱ 15 min

✓ Calculator OK

📘 Definition

Sum of the First n Terms

$S_{n}$ = sum of first n terms

There are two equivalent formulas for the sum of an arithmetic series, used in different scenarios:

Example:

$S_{n} = \frac{n}{2} (u_{1} + u_{n})$ (when you know first/last term)\n2. $S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d)$ (when you know $u_{1}$ and $d$ )

Arithmetic series are commonly used to model real-world scenarios, such as regular incremental savings, stacked objects, or monthly payments that increase by a constant amount each period.

📐 Worked Example

Liam saves $100 in January, $110 in February, $120 in March, and so on, increasing his monthly savings by $10 each month. What is his total savings after 1 year?

1
Identify the known values: 1 year = 12 months, so $n = 12$ . $u_{1} = 100$ , $d = 10$ .
2
Use the second sum formula since we know $u_{1}$ and $d$ :
3
$S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right)$
4
Substitute the values and calculate:
5
$S_{12} = \frac{12}{2}\left(2(100) + 11(10)\right) = 6\left(200 + 110\right) = 6(310) = 1860$
6
Liam's total savings after 1 year is $1860.

4. Common Pitfalls

Wrong move:

Using $n$ instead of $(n - 1)$ in the nth term or sum formula

Why:

You only add the common difference $(n - 1)$ times to reach the nth term starting from the first term

Correct move:

Always confirm by testing $n = 1$ ; if $u_{1}$ matches, the formula is set up correctly

Wrong move:

Calculating $d$ as positive when the sequence is decreasing

Why:

Students often subtract the previous term from the next term in reverse order

Correct move:

Always calculate $d = u_{n + 1} - u_{n}$ (next term minus previous term) to get the correct sign

Wrong move:

Miscounting the number of terms $n$ in word problems

Why:

For example, counting from month 1 to month 12 as 11 terms instead of 12

Correct move:

Always explicitly count the first term as $n = 1$ , and double-check the total number of terms before calculating

Wrong move:

Mixing up arithmetic and geometric sequence formulas

Why:

Confusing constant addition (arithmetic) with constant multiplication (geometric)

Correct move:

Check if the difference between terms is constant (arithmetic) or the ratio is constant (geometric) before applying a formula

5. Quick Reference Cheatsheet

Concept	Formula	Best For
Common difference	d = u_{n+1} - u_n	Checking if a sequence is arithmetic
nth term	u_n = u_1 + (n-1)d	Finding any term in the sequence
Sum (know u₁ + uₙ)	S_n = \frac{n}{2}(u_1 + u_n)	When first and last term are given
Sum (know u₁ + d)	S_n = \frac{n}{2}(2u_1 + (n-1)d)	When first term and difference are given

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find sum of first 15 terms
2022 · 2
Solve savings problem
2023 · 1
Find nth term given two terms

What's Next

Arithmetic sequences and series are the foundation for all sequence and series work in IB AA SL. The structure you learn here (identifying sequence types, finding terms, calculating sums) is very similar to geometric sequences, the next core sequence type you will study. Both are often tested together in exam questions, and you will use these concepts when learning sigma notation for writing sums of sequences concisely. Mastery of this sub-topic is essential for all further work on sequences, which makes up a significant portion of the Number & Algebra unit.