What is an Arithmetic Sequence?

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference, denoted $d$ .

In an arithmetic sequence $u_{1}, u_{2}, u_{3}, \dots$ , each term is obtained by adding $d$ to the previous one:

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

The common difference can be positive (increasing sequence), negative (decreasing sequence), or even zero (constant sequence).

Exam Tip

To find $d$ quickly, subtract any term from the one that follows it:
$d = u_{n + 1} - u_{n}$
This works for any consecutive pair in the sequence.

Note

You may see terms written as $a_{n}$ or $u_{n}$ — both notations mean exactly the same thing. IB exam papers typically use $u_{n}$ , so get comfortable with that.

Example

Consider the sequence: 3, 7, 11, 15, 19, …

First term: $u_{1} = 3$
Common difference: $d = 7 - 3 = 4$
Check: $11 - 7 = 4$ ✓, $15 - 11 = 4$ ✓

Each term is exactly 4 more than the previous one.

Free preview

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What is an Arithmetic Sequence?

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference, denoted $d$ .

In an arithmetic sequence $u_{1}, u_{2}, u_{3}, \dots$ , each term is obtained by adding $d$ to the previous one:

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

The common difference can be positive (increasing sequence), negative (decreasing sequence), or even zero (constant sequence).

Exam Tip

To find $d$ quickly, subtract any term from the one that follows it:
$d = u_{n + 1} - u_{n}$
This works for any consecutive pair in the sequence.

Note

You may see terms written as $a_{n}$ or $u_{n}$ — both notations mean exactly the same thing. IB exam papers typically use $u_{n}$ , so get comfortable with that.

Example

Consider the sequence: 3, 7, 11, 15, 19, …

First term: $u_{1} = 3$
Common difference: $d = 7 - 3 = 4$
Check: $11 - 7 = 4$ ✓, $15 - 11 = 4$ ✓

Each term is exactly 4 more than the previous one.

Free preview

SL 1.2—Sequences and Sigma Notation

What is an Arithmetic Sequence?

7 more sections in this topic

SL 1.2—Sequences and Sigma Notation

What is an Arithmetic Sequence?

7 more sections in this topic