Sequences

Numerical Sequence

In mathematics, a sequence is more than just a pattern; it's an ordered list of numbers where each number has a specific position (its place in the list). To work with sequences effectively, we use a special notation to distinguish a term's position from its value.

Definition Numerical Sequence

A numerical sequence is an ordered list of numbers, called terms. We use the notation $u_n$ to describe a term in the sequence.

The subscript $n$ is the index, which tells us the position of the term (often starting from $n=0$). The index $n$ usually takes integer values: $0,1,2,\dots$
$u_n$ represents the value of the term at that specific position.

So, $u_0$ is the value of the term at position $0$, $u_1$ is the value at position $1$, and so on.

Index ($n$)	0	1	2	$\dots$
Term ($u_n$)	$u_0$	$u_1$	$u_2$	$\dots$

Example

Given the sequence defined by the table below, find the value of the term $u_4$.

$n$	0	1	2	3	4	5	$\dots$
$u_n$	3	5	7	9	11	13	$\dots$

Answer

To find $u_4$, we look in the table for the column where the index is $n=4$. The value in the row below it is $11$.
Therefore, $u_4 = 11$.

Recursive Definition

Discover

Let’s consider a sequence where the first term is $2$, and each term is obtained by adding $3$ to the previous term. The terms are:

Here the sequence is indexed starting from $n=0$: $u_0 = 2$, $u_1 = 5$, $u_2 = 8$, $u_3 = 11$, etc.We can describe the relationship between the terms using formal notation:

$5 = 2\textcolor{colordef}{+3} \longrightarrow u_1 = u_0\textcolor{colordef}{+3} \longrightarrow u_{0+1} = u_0+3$
$8 = 5\textcolor{colordef}{+3} \longrightarrow u_2 = u_1\textcolor{colordef}{+3} \longrightarrow u_{1+1} = u_1+3$
$11 = 8\textcolor{colordef}{+3} \longrightarrow u_3 = u_2\textcolor{colordef}{+3} \longrightarrow u_{2+1} = u_2+3$

This pattern shows that any term $u_{n+1}$ can be found by adding $3$ to the previous term $u_n$. We can generalize this relationship as a rule (valid for all integers $n \ge 0$):$$u_{n+1} = u_n + 3$$

Definition Recursive Definition

A recursive definition of a sequence includes two parts:

The first term (or initial term), denoted for example by $u_0$ or $u_1$. This is the starting point.
The recursive rule (or recurrence relation), which is a formula that connects the next term, $u_{n+1}$, to the current term, $u_n$, for all appropriate integers $n$ (such as $n \ge 0$ or $n \ge 1$).

Once these two parts are known, every term in the sequence can be calculated step by step.

Example

A sequence is defined recursively by:

$u_1 = 5$ (The first term is $5$).
$u_{n+1} = u_n + 3$ (The rule is to add $3$ to the previous term, for all integers $n \ge 1$).

Find the first five terms of this sequence.

Answer

Let's build the sequence step-by-step using the recursive definition.

1$^{\text{st}}$ term: The starting term is given: $\boldsymbol{u_1 = 5}$.
2$^{\text{nd}}$ term: Use the rule with $n=1$. $\boldsymbol{u_2} = u_1 + 3 = 5 + 3 = 8$.
3$^{\text{rd}}$ term: Use the rule with $n=2$. $\boldsymbol{u_3} = u_2 + 3 = 8 + 3 = 11$.
4$^{\text{th}}$ term: Use the rule with $n=3$. $\boldsymbol{u_4} = u_3 + 3 = 11 + 3 = 14$.
5$^{\text{th}}$ term: Use the rule with $n=4$. $\boldsymbol{u_5} = u_4 + 3 = 14 + 3 = 17$.

$$5 \textcolor{olive}{\xrightarrow{\;+3\;}} 8 \textcolor{olive}{\xrightarrow{\;+3\;}} 11 \textcolor{olive}{\xrightarrow{\;+3\;}} 14 \textcolor{olive}{\xrightarrow{\,+3\,}} 17$$The first five terms are: $5, 8, 11, 14, 17$.

Arithmetic Sequence

An arithmetic sequence is the most common type of sequence that follows a recursive rule. It is defined by a starting term and a constant change between consecutive terms.

Definition Arithmetic Sequence

An arithmetic sequence is a sequence where each term after the first is found by adding a constant value to the previous term. This constant is called the common difference, denoted by $d$. It can be positive, negative, or zero.
The recursive definition of an arithmetic sequence is:$$u_{n+1} = u_n + d \quad \text{for all integers } n.$$

Example

An arithmetic sequence has a first term $u_1 = 5$ and a common difference $d = 3$.
Find the first five terms of this sequence.

Answer

The recursive rule is $u_{n+1} = u_n + 3$. We start with $u_1 = 5$ and apply the rule repeatedly.

$u_1 = \boldsymbol{5}$ (given)
$u_2 = u_1 + 3 = 5 + 3 = \boldsymbol{8}$
$u_3 = u_2 + 3 = 8 + 3 = \boldsymbol{11}$
$u_4 = u_3 + 3 = 11 + 3 = \boldsymbol{14}$
$u_5 = u_4 + 3 = 14 + 3 = \boldsymbol{17}$

The first five terms are: $5, 8, 11, 14, 17$.

Geometric Sequence

A geometric sequence is another fundamental type of sequence that follows a recursive rule. It is defined by a starting term and a constant multiplicative factor: to get from one term to the next, you always multiply by the same number.

Definition Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant non-zero value. This constant is called the common ratio, denoted by $r$.
The recursive definition of a geometric sequence is:$$u_{n+1} = r \times u_n \quad \text{for all integers } n.$$

Example

A geometric sequence has a first term $u_1 = 2$ and a common ratio $r = 2$.
Find the first five terms of this sequence.

Answer

The recursive rule is $u_{n+1} = u_n \times 2$. We start with $u_1 = 2$ and apply the rule repeatedly.

$u_1 = \boldsymbol{2}$ (given)
$u_2 = 2 \times u_1 = 2 \times 2 = \boldsymbol{4}$
$u_3 = 2 \times u_2 = 2 \times 4 = \boldsymbol{8}$
$u_4 = 2 \times u_3 = 2 \times 8 = \boldsymbol{16}$
$u_5 = 2 \times u_4 = 2 \times 16 = \boldsymbol{32}$

The first five terms are: $2, 4, 8, 16, 32$.

Index (\(n\))	0	1	2	\(\dots\)
Term (\(u_n\))	\(u_0\)	\(u_1\)	\(u_2\)	\(\dots\)

\(n\)	0	1	2	3	4	5	\(\dots\)
\(u_n\)	3	5	7	9	11	13	\(\dots\)