Sequences

Numerical Sequence

Definition Numerical Sequence

A numerical sequence is an ordered list of numbers $(u_0,\,u_1,\,u_2,\dots)$ defined by a rule.

$n$	0	1	2	$\dots$
$u_n$	$u_0$	$u_1$	$u_2$	$\dots$

The number $u_n$ is called the $n$th term of the sequence.

Example

What is $u_4$ of this sequence?

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

Answer

$u_4 = 11$.

Definition Using a Recursive Rule

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:

We observe:

$5 = 2\textcolor{colordef}{+3} \longrightarrow u_1 = u_0\textcolor{colordef}{+3} \longrightarrow u_{0+1} = u_0+3$
$8 = 5\textcolor{colordef}{+3} \longrightarrow u_2 = u_1\textcolor{colordef}{+3} \longrightarrow u_{1+1} = u_1+3$
$11 = 8\textcolor{colordef}{+3} \longrightarrow u_3 = u_2\textcolor{colordef}{+3} \longrightarrow u_{2+1} = u_2+3$
$\vdots$
So the rule is $u_{n+1} = u_n + 3$

Definition Recursive Rule

A sequence can be defined by:

the first term (starting number): $u_0$
a recursive rule that tells how to obtain each term from the previous one:$u_{n+1}=$ expression in $u_n$

Example

Write the recursive rule when each term is obtained by adding $2$ to the previous term.

Answer

$$u_{n+1}=u_n+2$$

Definition Using an Explicit Rule

Definition Explicit Rule

A sequence can also be defined by an explicit rule (or explicit formula), which gives a direct formula for the $n$th term in terms of $n$:$$u_n = \text{expression in } n$$

Example

Consider the sequence defined by the explicit formula: $u_n = 3n + 2$.
Write the first five terms of this sequence.

Answer

For $n=0$: $$ \begin{aligned}[t] u_0 &= 3 \times 0 + 2 \\ &= 0 + 2 \\ &= 2 \end{aligned} $$
For $n=1$: $$ \begin{aligned}[t] u_1 &= 3 \times 1 + 2 \\ &= 3 + 2 \\ &= 5 \end{aligned} $$
For $n=2$: $$ \begin{aligned}[t] u_2 &= 3 \times 2 + 2 \\ &= 6 + 2 \\ &= 8 \end{aligned} $$
For $n=3$: $$ \begin{aligned}[t] u_3 &= 3 \times 3 + 2 \\ &= 9 + 2 \\ &= 11 \end{aligned} $$
For $n=4$: $$ \begin{aligned}[t] u_4 &= 3 \times 4 + 2 \\ &= 12 + 2 \\ &= 14 \end{aligned} $$

So the first five terms are: $2,\ 5,\ 8,\ 11,\ 14$.

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

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