What is an Arithmetic Sequence?

Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is constant. This constant value 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:

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

Some key vocabulary:

$u_{1}$ (or $a_{1}$ ) , the first term
$d$ , the common difference (can be positive, negative, or zero)
$u_{n}$ , the $n$ th term (the general term)

Note

The notation $a_{n}$ and $u_{n}$ are completely interchangeable. IB exams typically use $u_{n}$ , so we'll stick with that , but don't be thrown if you see $a_{n}$ in other resources.

Example

Consider the sequence: $3, 7, 11, 15, 19, \dots$

First term: $u_{1} = 3$
Common difference: $d = 7 - 3 = 4$
Each term is found by adding 4 to the previous one.

Now try: $20, 15, 10, 5, 0, \dots$

First term: $u_{1} = 20$
Common difference: $d = 15 - 20 = - 5$ (negative $d$ means the sequence is decreasing)

Exam Tip

To find $d$ quickly, subtract any term from the term that immediately follows it: $d = u_{n + 1} - u_{n}$
Do this check with two different pairs , if $d$ is the same both times, you've confirmed it's arithmetic.

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What is an Arithmetic Sequence?

Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is constant. This constant value 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:

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

Some key vocabulary:

$u_{1}$ (or $a_{1}$ ) , the first term
$d$ , the common difference (can be positive, negative, or zero)
$u_{n}$ , the $n$ th term (the general term)

Note

The notation $a_{n}$ and $u_{n}$ are completely interchangeable. IB exams typically use $u_{n}$ , so we'll stick with that , but don't be thrown if you see $a_{n}$ in other resources.

Example

Consider the sequence: $3, 7, 11, 15, 19, \dots$

First term: $u_{1} = 3$
Common difference: $d = 7 - 3 = 4$
Each term is found by adding 4 to the previous one.

Now try: $20, 15, 10, 5, 0, \dots$

First term: $u_{1} = 20$
Common difference: $d = 15 - 20 = - 5$ (negative $d$ means the sequence is decreasing)

Exam Tip

Free preview

SL 1.2—Arithmetic sequences and series

What is an Arithmetic Sequence?

9 more sections in this topic

SL 1.2—Arithmetic sequences and series

What is an Arithmetic Sequence?

9 more sections in this topic