What Is a Sequence?

A sequence is an ordered list of numbers that follows a particular rule or pattern.

For example:

2, 4, 6, 8, 10, ...

is a sequence because the numbers follow a clear pattern:

Add 2 each time

The order of the numbers matters. Changing the order changes the sequence.

Terms of a Sequence

Each number in a sequence is called a term.

Consider:

5, 8, 11, 14, 17, ...

The terms are:

• first term = 5
• second term = 8
• third term = 11
• fourth term = 14
• fifth term = 17

We can refer to the position of each term using notation.

Sequence Notation

A common way to represent the terms of a sequence is:

\( a_1, a_2, a_3, a_4, ... \)

where:

• a₁ = first term
• a₂ = second term
• a₃ = third term
• a_n = the n^th term

For the sequence:

3, 7, 11, 15, ...

we have:

a₁ = 3

a₂ = 7

a₃ = 11

a₄ = 15

What Does a_n Mean?

The notation a_n means the term in position n.

For example: a₅ means the fifth term.

If a sequence is: 10, 20, 30, 40, 50, ...

then: a₅ = 50

The subscript tells us the position, not the value.

Ordered Lists

A sequence is an ordered list.

Compare: 2, 4, 6, 8

and: 8, 6, 4, 2

They contain the same four numbers, but they are different sequences because the numbers appear in a different order.

This is important because sequences often describe how something changes from one step to the next.

Identifying Patterns

Many sequences follow simple numerical patterns.

For example: 4, 7, 10, 13, 16, ...

The pattern is: +3

Each term is found by adding 3 to the previous term.

Example: Subtraction Pattern

Consider: 30, 25, 20, 15, 10, ...

The pattern is: -5

Each term decreases by 5.

The next two terms are: 5, 0

Example: Multiplication Pattern

Consider: 2, 6, 18, 54, ...

Each term is multiplied by: 3

So the pattern is: x3

The next term is: 5x3 = 162

Example: Division Pattern

Consider: 160, 80, 40, 20, 10, ...

Each term is divided by: 2

So: ÷ 2

The next term is: 5

Not All Patterns Are Constant

Some sequences use more complicated patterns.

For example: 1, 4, 9, 16, 25, ...

These numbers are: \( 1^2, 2, ^2, 3^2, 4^2, 5^2, ... \)

This is the sequence of square numbers.

The difference between terms is not constant:

but there is still a clear pattern.

Another Example: Increasing Differences

Consider: 2, 5, 9, 14, 20, ...

Look at the differences: +3, +4, +5, +6

The amount added increases by 1 each time.

Therefore, the next difference is: +7

and the next term is: 20 + 7 = 27

Fibonacci-Type Sequences

Some sequences are created using earlier terms.

One famous example is:

1, 1, 2, 3, 5, 8, 13, ...

Each term is found by adding the previous two terms.

For example:

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

This is called the Fibonacci sequence.

Sequence or Random List?

A sequence should follow an identifiable rule or structure.

Consider: 4, 8, 12, 16, ...

This is clearly a sequence because: +4 is repeated.

Now consider: 4, 17, 2, 39, 11

If no rule or relationship is given, this may simply be a random list of numbers.

However, we must be careful: sometimes a complicated sequence may not have an obvious pattern at first.

The important idea is that a mathematical sequence is generated according to some defined rule.

Finite and Infinite Sequences

Sequences can be finite or infinite.

Finite Sequence

A finite sequence ends.

For example: 2, 4, 6, 8, 10

There are exactly five terms.

Infinite Sequence

An infinite sequence continues without ending.

For example: 2, 4, 6, 8, 10, ...

The dots: ... show that the pattern continues.

Describing a Sequence in Words

Sequences can often be described using words.

For example: 6, 10, 14, 18, ...

can be described as:

Start at 6 and add 4 each time.

Another sequence: 100, 50, 25, 12.5, ...

can be described as:

Start at 100 and divide by 2 each time.

Being able to describe the pattern is an important step toward finding a mathematical formula.

Using a Table

Sequences can also be represented using a table.

Consider: 3, 6, 9, 12, 15, ...

This table makes the relationship between the term number and the term value easier to see.

The n^th Term

Instead of listing every term, we can sometimes write a formula that gives any term directly.

For the sequence: 3, 6, 9, 12, ...

the pattern is: a_n = 3n

Check:

For n = 1:

a₁ = 3(1) = 3

For n = 4:

a₄ = 3(4) = 12

For n = 10:

a₁₀ = 3(10) = 30

The formula allows us to find any term without writing all the earlier terms.

Worked Example: Using Sequence Notation

Suppose: a_n = 4n + 1

Find the first four terms.

For n = 1:

a₁ = 4(1) + 1 = 5

For n = 2: 

a₂ = 4(2) + 1 = 9

For n = 3:

a₃ = 4(3) + 1 = 13

For n = 4:

a₄ = 4(4) + 1 = 17

Therefore, the sequence is:

5, 9, 13, 17, ...

Worked Example: Finding a Specific Term

Suppose:

a_n = 7n - 2

Find the sixth term.

Substitute:

n = 6

a₆ = 7(6) - 2

a₆ = 42 - 2

a₆ = 40

Connecting Sequences to Graphs

A sequence can also be represented graphically.

Consider:

a_n = 2n + 1

The first few terms are:

3, 5, 7, 9, 11, ...

These can be plotted as:

(1, 3), (2, 5), (3, 7), (4, 9), (5, 11)

The horizontal axis represents: n

and the vertical axis represents: a_n

Unlike a continuous line graph, sequence graphs normally contain separate points because n takes integer values such as: 1, 2, 3, 4, ...

Sequences in the Real World

Sequences appear in many real situations.

Saving Money

Suppose you save an additional $20 each week:

20, 40, 60, 80, ...

Rows of Seats

A theatre might have:

15, 17, 19, 21, ...

seats in successive rows.

Population Growth

A population might double:

100, 200, 400, 800, ...

Bouncing Ball

The height of a bouncing ball may decrease according to a repeated multiplier.

Sequences are therefore useful for modelling change over time or position.

Recognising Different Types of Patterns

When examining a sequence, ask:

Is the same number being added or subtracted?

Example: 5, 9, 13, 17

Difference: +4

Is the same number being multiplied or divided?

Example: 3, 6, 12, 24

Multiplier: x2

Are the differences themselves changing regularly?

Example: 1, 4, 9, 16

Differences: 3, 5, 7

Does each term depend on previous terms?

Example: 1, 1, 2, 3, 5, 8

These questions can help identify the underlying rule.

Worked Example: Identify the Pattern

Consider: 7, 12, 17, 22, 27, ...

Find the difference between consecutive terms:

12 - 7 = 5

17 - 12 = 5

22 - 17 = 5

Therefore: Add 5 each time

The next term is: 27 + 5 = 32

Worked Example: A Less Obvious Pattern

Consider: 2, 6, 12, 20, 30

Look at the differences: +4, +6, +8, + 10

The differences increase by 2.

The next difference should therefore be: + 12

So: 30 + 12 = 42

The next term is: 42

Common Misconceptions

The term number and the term value are not the same thing.

In: 4, 8, 12, 16

the third term is: a₃ = 12

The 3 tells us the position, while 12 is the value.

A pattern does not always involve addition.

Sequences can involve:

• subtraction
• multiplication
• division
• powers
• previous terms
• changing differences

A sequence does not need to increase.

For example: 20, 15, 10, 5, 0 is still a sequence.

A sequence does not need to contain positive integers.

For example: 2.5, 2, 1.5, 1, 0.5, ... is a valid sequence.

Did You Know?

Sequences have been studied for thousands of years.

They are important in:

• algebra
• geometry
• finance
• computer science
• population modelling
• physics
• probability

The study of sequences also leads naturally into series, where the terms of a sequence are added together.

For example, from the sequence: 1, 2, 3, 4, ...

we can form the series: 1 + 2 + 3 + 4 + ...

Key Terms

Sequence – An ordered list of numbers generated according to a rule.

Term – An individual number in a sequence.

Term number – The position of a term in the sequence. a_n – Notation representing the n^th term of a sequence.

Pattern – A consistent relationship between terms.

Finite sequence – A sequence containing a fixed number of terms.

Infinite sequence – A sequence that continues indefinitely.

n^th-term rule – A formula used to calculate any term from its position.

Key Takeaways

• A sequence is an ordered list of numbers that follows a rule or pattern.
• Each number in a sequence is called a term.
• The notation: a_n represents the n^th term.
• The order of terms matters.
• Patterns may involve addition, subtraction, multiplication, division, powers, or relationships between previous terms.
• Sequences may be finite or infinite.
• An n^th-term formula can be used to calculate a term directly.
• A sequence can be represented using a list, table, formula, or graph.
• Sequence graphs usually consist of separate points because the term number n normally takes positive integer values.
• Recognising patterns in sequences provides the foundation for studying arithmetic sequences, geometric sequences, recursive sequences, and series.