1.7.2 Types of sequence (A-level only)

What you'll learn

How to generate a sequence from an nth-term formula or a recurrence relation.
How to decide whether a sequence is increasing or decreasing.
How to recognise and describe periodic sequences.
How to justify a sequence's type using algebra rather than checking only a few terms.

Sequences and their notation

A sequence is an ordered list of numbers called terms. The position of each term matters.

For example, the sequence

15,…3,\ 7,\ 11,\ 15,\ldots

has first term 3, second term 7 and so on.

Definition

Term notation

The symbol $x_n$ means the term in position $n$ . Thus, $x_1$ is the first term, $x_2$ is the second term and $x_{10}$ is the tenth term.

The letter used does not matter: you may also see $u_n$ , $a_n$ or another letter.

A sequence is made of separate terms, usually corresponding to positive integer values of $n$ . If its terms are plotted, the points should not be treated as a continuous curve.

Three term-number plots showing increasing, decreasing and periodic sequences

Sequences defined by an nth term

An nth-term formula gives each term directly in terms of its position $n$ . To find a particular term, substitute its position into the formula.

For example, if

x_n=3n^2-2,

then any term can be calculated without finding the terms before it.

Example

Generating terms from an nth-term formula

The sequence is defined by $x_n=3n^2-2$ . Find its first three terms and its tenth term.

Substitute $n = 1$ , $n = 2$ and $n = 3$ :
$x1=3(1)2−2=1,x2=3(2)2−2=10,x3=3(3)2−2=25.\begin{aligned} x_1&=3(1)^2-2=1,\\ x_2&=3(2)^2-2=10,\\ x_3&=3(3)^2-2=25. \end{aligned}$
Therefore, the sequence begins
$25,…1,\ 10,\ 25,\ldots$
Substitute $n = 10$ directly to find the tenth term:
$x_{10}=3(10)^2-2=298.$

Sequences defined recursively

A recurrence relation defines each new term using one or more earlier terms.

A simple recurrence has the form

x_{n+1}=f(x_n),

where $f$ is a function. This means that the next term is found by applying the rule $f$ to the current term.

Key Idea

A starting value is essential

A recurrence relation does not usually define a unique sequence by itself. You also need an initial value, such as $x_1=4$ , to start the process.

Example

Generating terms recursively

A sequence is defined by

x1=4,xn+1=3xn−2.x_1=4,\qquad x_{n+1}=3x_n-2.

Find the next three terms.

Apply the recurrence to $x_1=4$ :
$x_2=3x_1-2=3(4)-2=10.$
Use the newly calculated value to find the next term:
$x_3=3x_2-2=3(10)-2=28.$
Repeat the process once more:
$x_4=3x_3-2=3(28)-2=82.$
The sequence begins $82,…4,\ 10,\ 28,\ 82,\ldots$

Common Mistake

Substituting the term number

In $x_{n+1}=3x_n-2$ , the input is the previous term value, not the term number. To find $x_4$ , use $x_3$ , not $n = 3$ .

Increasing sequences

An increasing sequence has terms that get larger as $n$ increases.

Definition

Strictly increasing sequence

A sequence is strictly increasing if

x_{n+1}>x_n

for every permitted value of $n$ .

A sequence is non-decreasing if $xn+1≥xnx_{n+1}\ge x_n$ . This allows consecutive terms to be equal. Questions sometimes use “increasing” for this weaker meaning, so read the wording carefully.

Checking several terms can suggest that a sequence is increasing, but it does not prove that every later term continues to increase.

For a sequence defined by an nth term, examine the difference

x_{n+1}-x_n.

If this difference is positive for every permitted $n$ , the sequence is strictly increasing.
If it is non-negative, the sequence is non-decreasing.

Decreasing sequences

A decreasing sequence has terms that get smaller as $n$ increases.

Definition

Strictly decreasing sequence

A sequence is strictly decreasing if

x_{n+1}<x_n

for every permitted value of $n$ .

Similarly, a sequence is non-increasing if $xn+1≤xnx_{n+1}\le x_n$ , allowing equal consecutive terms.

The same difference test works:

$x_{n+1}-x_n<0$ proves that the sequence is strictly decreasing.
$xn+1−xn≤0x_{n+1}-x_n\le 0$ proves that it is non-increasing.

Example

Classifying a sequence using differences

Determine whether the sequence $x_n=n^2-4n$ is increasing or decreasing for $n≥1n\ge1$ .

Form the next term by replacing $n$ with $n + 1$ :
$x_{n+1}=(n+1)^2-4(n+1)=n^2-2n-3.$
Subtract the current term:
$xn+1−xn=(n2−2n−3)−(n2−4n)=2n−3.\begin{aligned} x_{n+1}-x_n &=(n^2-2n-3)-(n^2-4n)\\ &=2n-3. \end{aligned}$
Test the sign over the required domain. At $n = 1$ , the difference is negative, but for $n≥2n\ge2$ it is positive. Therefore, the sequence decreases from $x_1$ to $x_2$ , then increases from that point onwards. It is neither increasing nor decreasing over the whole domain $n≥1n\ge1$ .

Tip

Use the sign, not the size of the terms

An increasing sequence can contain negative terms, and a decreasing sequence can contain positive terms. Compare consecutive terms rather than asking whether the terms themselves are positive or negative.

For a recurrence relation, it may be possible to rearrange the rule to identify the sign of $x_{n+1}-x_n$ directly. However, take care: the sign may depend on the current value $x_n$ .

Periodic sequences

A periodic sequence repeats after a fixed number of terms.

Definition

Period

A sequence has period $p$ if $p$ is a positive integer such that

x_{n+p}=x_n

for every permitted value of $n$ . The smallest possible positive value of $p$ is called the fundamental period.

For example, the sequence

5,…2,\ 5,\ 2,\ 5,\ 2,\ 5,\ldots

has fundamental period 2.

A constant sequence has period 1 because every term is the same.

Example

Identifying a periodic recurrence

A sequence is defined by

x1=3,xn+1=1−xn.x_1=3,\qquad x_{n+1}=1-x_n.

Show that it is periodic and state its fundamental period.

Generate the next terms:
$x2=1−3=−2,x3=1−(−2)=3,x4=1−3=−2.\begin{aligned} x_2&=1-3=-2,\\ x_3&=1-(-2)=3,\\ x_4&=1-3=-2. \end{aligned}$
This suggests the repeating pattern $−2,…3,\ -2,\ 3,\ -2,\ldots$
Apply the recurrence twice to justify the repetition:
$x_{n+2}=1-x_{n+1}=1-(1-x_n)=x_n.$
Therefore, every term repeats two places later.
Since the sequence is not constant, its period cannot be 1. Its fundamental period is therefore 2.

A sequence may also be periodic when defined by an nth term. For instance,

xn=cos⁡(nπ2)x_n=\cos\left(\frac{n\pi}{2}\right)

generates $1,…0,\ -1,\ 0,\ 1,\ldots$ and satisfies $x_{n+4}=x_n$ , so its fundamental period is 4.

Common Mistake

Assuming an observed pattern is proof

Writing down a few repeated terms provides evidence of periodicity, but an algebraic argument such as $x_{n+p}=x_n$ proves that the pattern continues indefinitely.

Sequences that do not fit one type

Not every sequence is increasing, decreasing or periodic.

For example, $x_n=(-1)^n n$ generates

−5,…-1,\ 2,\ -3,\ 4,\ -5,\ldots

The terms alternate in sign while their magnitudes grow. The sequence is neither increasing nor decreasing, and it does not repeat, so it is not periodic.

Exam technique

In the exam

Identify whether you have an nth-term formula or a recurrence relation; for a recurrence, locate the starting value.
To test increasing or decreasing behaviour, calculate and simplify $x_{n+1}-x_n$ , then justify its sign over the stated values of $n$ .
To prove periodicity, show that $x_{n+p}=x_n$ for every relevant $n$ , and check whether a smaller positive period is possible.
Pay attention to the starting index and to whether the question requires strict or non-strict inequalities.

Self review

Check yourself

How would you prove that $x_n=2n^2+3$ is strictly increasing for $n≥1n\ge1$ ?
Generate the first five terms of $x_1=2$ , $x_{n+1}=4-x_n$ , and identify the fundamental period.
Can a sequence be non-decreasing and non-increasing at the same time? What would such a sequence look like?

What you'll learn

How to generate a sequence from an nth-term formula or a recurrence relation.
How to decide whether a sequence is increasing or decreasing.
How to recognise and describe periodic sequences.
How to justify a sequence's type using algebra rather than checking only a few terms.

Sequences and their notation

A sequence is an ordered list of numbers called terms. The position of each term matters.

For example, the sequence

15,…3,\ 7,\ 11,\ 15,\ldots

has first term 3, second term 7 and so on.

Definition

Term notation

The symbol $x_n$ means the term in position $n$ . Thus, $x_1$ is the first term, $x_2$ is the second term and $x_{10}$ is the tenth term.

The letter used does not matter: you may also see $u_n$ , $a_n$ or another letter.

A sequence is made of separate terms, usually corresponding to positive integer values of $n$ . If its terms are plotted, the points should not be treated as a continuous curve.

Three term-number plots showing increasing, decreasing and periodic sequences

Sequences defined by an nth term

An nth-term formula gives each term directly in terms of its position $n$ . To find a particular term, substitute its position into the formula.

For example, if

x_n=3n^2-2,

then any term can be calculated without finding the terms before it.

Example

Generating terms from an nth-term formula

The sequence is defined by $x_n=3n^2-2$ . Find its first three terms and its tenth term.

Substitute $n = 1$ , $n = 2$ and $n = 3$ :
$x1=3(1)2−2=1,x2=3(2)2−2=10,x3=3(3)2−2=25.\begin{aligned} x_1&=3(1)^2-2=1,\\ x_2&=3(2)^2-2=10,\\ x_3&=3(3)^2-2=25. \end{aligned}$
Therefore, the sequence begins
$25,…1,\ 10,\ 25,\ldots$
Substitute $n = 10$ directly to find the tenth term:
$x_{10}=3(10)^2-2=298.$

Sequences defined recursively

A recurrence relation defines each new term using one or more earlier terms.

A simple recurrence has the form

x_{n+1}=f(x_n),

where $f$ is a function. This means that the next term is found by applying the rule $f$ to the current term.

Key Idea

A starting value is essential

A recurrence relation does not usually define a unique sequence by itself. You also need an initial value, such as $x_1=4$ , to start the process.

Example

Generating terms recursively

A sequence is defined by

x1=4,xn+1=3xn−2.x_1=4,\qquad x_{n+1}=3x_n-2.

Find the next three terms.

Apply the recurrence to $x_1=4$ :
$x_2=3x_1-2=3(4)-2=10.$
Use the newly calculated value to find the next term:
$x_3=3x_2-2=3(10)-2=28.$
Repeat the process once more:
$x_4=3x_3-2=3(28)-2=82.$
The sequence begins $82,…4,\ 10,\ 28,\ 82,\ldots$

Common Mistake

Substituting the term number

In $x_{n+1}=3x_n-2$ , the input is the previous term value, not the term number. To find $x_4$ , use $x_3$ , not $n = 3$ .

Increasing sequences

An increasing sequence has terms that get larger as $n$ increases.

Definition

Strictly increasing sequence

A sequence is strictly increasing if

x_{n+1}>x_n

for every permitted value of $n$ .

Checking several terms can suggest that a sequence is increasing, but it does not prove that every later term continues to increase.

For a sequence defined by an nth term, examine the difference

x_{n+1}-x_n.

If this difference is positive for every permitted $n$ , the sequence is strictly increasing.
If it is non-negative, the sequence is non-decreasing.

Decreasing sequences

A decreasing sequence has terms that get smaller as $n$ increases.

Definition

Strictly decreasing sequence

A sequence is strictly decreasing if

x_{n+1}<x_n

for every permitted value of $n$ .

Similarly, a sequence is non-increasing if $xn+1≤xnx_{n+1}\le x_n$ , allowing equal consecutive terms.

The same difference test works:

$x_{n+1}-x_n<0$ proves that the sequence is strictly decreasing.
$xn+1−xn≤0x_{n+1}-x_n\le 0$ proves that it is non-increasing.

Example

Classifying a sequence using differences

Determine whether the sequence $x_n=n^2-4n$ is increasing or decreasing for $n≥1n\ge1$ .

Form the next term by replacing $n$ with $n + 1$ :
$x_{n+1}=(n+1)^2-4(n+1)=n^2-2n-3.$
Subtract the current term:
$xn+1−xn=(n2−2n−3)−(n2−4n)=2n−3.\begin{aligned} x_{n+1}-x_n &=(n^2-2n-3)-(n^2-4n)\\ &=2n-3. \end{aligned}$
Test the sign over the required domain. At $n = 1$ , the difference is negative, but for $n≥2n\ge2$ it is positive. Therefore, the sequence decreases from $x_1$ to $x_2$ , then increases from that point onwards. It is neither increasing nor decreasing over the whole domain $n≥1n\ge1$ .

Tip

Use the sign, not the size of the terms

For a recurrence relation, it may be possible to rearrange the rule to identify the sign of $x_{n+1}-x_n$ directly. However, take care: the sign may depend on the current value $x_n$ .

Periodic sequences

A periodic sequence repeats after a fixed number of terms.

Definition

Period

A sequence has period $p$ if $p$ is a positive integer such that

x_{n+p}=x_n

for every permitted value of $n$ . The smallest possible positive value of $p$ is called the fundamental period.

For example, the sequence

5,…2,\ 5,\ 2,\ 5,\ 2,\ 5,\ldots

has fundamental period 2.

A constant sequence has period 1 because every term is the same.

Example

Identifying a periodic recurrence

A sequence is defined by

x1=3,xn+1=1−xn.x_1=3,\qquad x_{n+1}=1-x_n.

Show that it is periodic and state its fundamental period.

Generate the next terms:
$x2=1−3=−2,x3=1−(−2)=3,x4=1−3=−2.\begin{aligned} x_2&=1-3=-2,\\ x_3&=1-(-2)=3,\\ x_4&=1-3=-2. \end{aligned}$
This suggests the repeating pattern $−2,…3,\ -2,\ 3,\ -2,\ldots$
Apply the recurrence twice to justify the repetition:
$x_{n+2}=1-x_{n+1}=1-(1-x_n)=x_n.$
Therefore, every term repeats two places later.
Since the sequence is not constant, its period cannot be 1. Its fundamental period is therefore 2.

A sequence may also be periodic when defined by an nth term. For instance,

xn=cos⁡(nπ2)x_n=\cos\left(\frac{n\pi}{2}\right)

generates $1,…0,\ -1,\ 0,\ 1,\ldots$ and satisfies $x_{n+4}=x_n$ , so its fundamental period is 4.

Common Mistake

Assuming an observed pattern is proof

Writing down a few repeated terms provides evidence of periodicity, but an algebraic argument such as $x_{n+p}=x_n$ proves that the pattern continues indefinitely.

Sequences that do not fit one type

Not every sequence is increasing, decreasing or periodic.

For example, $x_n=(-1)^n n$ generates

−5,…-1,\ 2,\ -3,\ 4,\ -5,\ldots

The terms alternate in sign while their magnitudes grow. The sequence is neither increasing nor decreasing, and it does not repeat, so it is not periodic.

Exam technique

In the exam

Identify whether you have an nth-term formula or a recurrence relation; for a recurrence, locate the starting value.
To test increasing or decreasing behaviour, calculate and simplify $x_{n+1}-x_n$ , then justify its sign over the stated values of $n$ .
To prove periodicity, show that $x_{n+p}=x_n$ for every relevant $n$ , and check whether a smaller positive period is possible.
Pay attention to the starting index and to whether the question requires strict or non-strict inequalities.

Self review

Check yourself

How would you prove that $x_n=2n^2+3$ is strictly increasing for $n≥1n\ge1$ ?
Generate the first five terms of $x_1=2$ , $x_{n+1}=4-x_n$ , and identify the fundamental period.
Can a sequence be non-decreasing and non-increasing at the same time? What would such a sequence look like?

What you'll learn

Sequences and their notation

Sequences defined by an nth term

Sequences defined recursively

Increasing sequences

Decreasing sequences

Periodic sequences

Sequences that do not fit one type

Review 1.7.2 Types of sequence (A-level only) by teaching Genie

1.7.2 Types of sequence (A-level only)

What you'll learn

Sequences and their notation

Sequences defined by an nth term

Sequences defined recursively

Increasing sequences

Decreasing sequences

Periodic sequences

Sequences that do not fit one type

Review 1.7.2 Types of sequence (A-level only) by teaching Genie

1.7.2 Types of sequence (A-level only)

What you'll learn

Sequences and their notation

Sequences defined by an nth term

Sequences defined recursively

Increasing sequences

Decreasing sequences

Periodic sequences

Sequences that do not fit one type

Review 1.7.2 Types of sequence (A-level only) by teaching Genie

1.7.2 Types of sequence (A-level only) Revision Guide

1.7.2 Types of sequence (A-level only)

What you'll learn

Sequences and their notation

Sequences defined by an nth term

Sequences defined recursively

Increasing sequences

Decreasing sequences

Periodic sequences

Sequences that do not fit one type

Review 1.7.2 Types of sequence (A-level only) by teaching Genie

1.7.2 Types of sequence (A-level only) Revision Guide