Introduction to Sequences - Mathematics.lu

A sequence is an ordered list of numbers or other objects. The order matters, and each item in the sequence is called a term. In this section, we learn how to define sequences formally, different types of sequences, and some of their properties.

Table of Contents

Motivation

Consider the following list of numbers:

$\{2,4,6,8,10,12,14,...\}$

What can we say about this sequence?

All its terms are even numbers.
Each term is obtained by adding $+2$ to the previous term.
Terms in the sequence get larger and larger as we continue through the sequence.
The sequence is infinite.

All of these properties can be defined formally and studying them can reveal interesting phenomena.

Basic Definition

Definition (Sequence): A sequence is an ordered list of numbers. It is conventionally written as a list enclosed in braces:

$\{u_1, u_2, u_3, \dots\}$

A sequence can be compactly represented using the index notation:

$\{u_n\}_{n=1}^{\infty}$

where:

$u_1$ : The absolute first term of the ordered list.
$u_n$ : The general term (or $n$ -th term) located at position index $n$ .
$n$ : The position index parameter ( $n = 1, 2, 3, \dots$ ).

We can also have sequences where the first term is $u_0$ , the second term is $u_1$ and so on. These are all analogous. Since the parameter $n$ describes a position, it must be a non-negative integer. We do not have to use the letter $u$ for sequences, they could also be $a_n,b_n,f_n$ and any kind.

Defining a Sequence Explicitly

Sequences do not necessarily have a general rule or function. However, sometimes they could have. The sequence we showed above of even numbers can be defined as $u_n=2n$ . Therefore, by replacing different values of $n$ we obtain different terms.

$u_1=2\cdot1=2, u_2=2\cdot2=4,u_3=2\cdot3=6,...$

Thus creating the sequence $\{2,4,6,8,10,...\}$ .

In this case, the sequence is said to be defined explicitly. This is because there is a general rule to describe the sequence.

Not every sequence can be defined explicitly using a closed-form formula. For example, we do not have a formula to describe the sequence of all prime numbers:

$\{2,3,5,7,11,13,17,19,...\}$

However, we know that there are infinite primes of the from $4n+1$ (for example, 5, 13, 17, 29, and infinite more), even though it should be clear that not every prime has this form.

Finding the explicit formula is not always easy, or it may even take some trial-and-error. Nonetheless, you are expected to be able to find terms of a sequence defined explicitly and understand other properties which we will discuss in future pages.

Recursive Sequences

A recursive sequence is one where some terms are given, and all the other terms are given by a rule. One of the most famous examples of a recursive sequence is the Fibonacci sequence.

Example: The Fibonacci sequence is a classic recursive sequence where each term is the sum of the two preceding terms. It is defined by the initial conditions $F_1 = 1$ , $F_2 = 1$ , and the recurrence relation rule:

$F_{n+2} = F_{n+1} + F_n \quad \text{for } n \ge 1$

Show the first 10 terms of the sequence, and demonstrate how terms $F_3$ , $F_4$ , and $F_5$ are explicitly obtained using the recursive rule.

We see that the sequence is defined as follows: The first and second term are 1. Every other term is obtained by adding the two terms before it. Another way to write this (using function notation) is:

$F_n=\begin{cases}1\quad\text{ if }n=1,2\\F_{n-2}+F_{n-1}\quad\text{ otherwise}\end{cases}$

Here we are saying that to find $F_5$ , for example, we have $n=5$ so we go to our rule “otherwise”. Here we are told that $F_5=F_4+F_3$ . This type of notation is easy to get used to, and is very common to describe recursive sequences.

Definition (Recursive Sequence): A recursive sequence is a sequence $\{u_n\}_{n=1}^{\infty}$ where subsequent terms are generated using a rule or formula that relies on the values of one or more preceding terms. To completely define a recursive sequence, two distinct structural parameters must be explicitly stated:

1. The Initial Conditions

The starting boundary values that seed the list matrix. This must include at least the first term $u_1$ , along with any other starting terms required by the specific formula rules.

2. The Recurrence Relation

An algebraic formula that defines the next term $u_{n+1}$ (or $u_n$ ) as a function of the current term $u_n$ (or past terms like $u_{n-1}$ ):

$u_{n+1} = f(u_n) \quad \text{for } n \ge 1$

By combining the initial terms with the recurrence rule, the list $\{u_1, u_2, u_3, \dots\}$ is generated sequentially, step by step.

Let us see another example of recursive sequences and how we find some of their terms using its definition.

Example: A first-order recursive sequence is defined by the initial condition $u_1 = 5$ and the recurrence relation rule:

$u_{n+1} = \frac{u_n - 3}{2} \quad \text{for } n \ge 1$

Find the exact values of the first 5 terms of this sequence, expressing any fractional results in simplified forms.

The Collatz Sequence

The Collatz sequence is generated from any positive integer by repeatedly applying these rules:

If the number is even, divide it by 2.
If the number is odd, multiply it by 3 and add 1.

This process is repeated with each new number.

For example, starting with 6:

$6\to3\to10\to5\to16\to8\to4\to2\to1$

Once you reach 1, the sequence enters the cycle: $1\to4\to2\to1\to...$ .

Formally, it can be defined as:

Definition (Collatz Sequence): Starting with any chosen positive integer $u_1$ , each subsequent term is calculated from the previous term using the function $f(n)$ defined as:

$f(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases}$

The recurrence relation to generate the next term in the sequence $\{u_n\}_{n=1}^{\infty}$ is given by:

$u_{n+1} = f(u_n)$

The Collatz Conjecture asserts that no matter what positive integer is chosen for the first term $u_1$ , the sequence will always eventually reach the number $1$ , trapping itself in an infinite repeating cycle of ${4, 2, 1}$ . Try it a few times with any positive integer to see that you end up in the same loop.

Optional – Arithmetic and Geometric Sequences

These sequences you examine more closely in S7. We can still discuss a few examples since they are relatively easy concepts to understand.

An arithmetic sequence is a sequence where every term is obtained from the previous one by adding a constant term called the difference. The sequence given at the beginning of this section $\{2,4,6,8,10,...\}$ is an example of an arithmetic sequence because we can write it as: $u_n=u_{n-1}+2$ . That is, every term is the result of the previous one being added the number 2. The difference between any two consecutive terms is always $2$ . E.g.: $4-2=6-4=8-6=...$ .

The general formula for an arithmetic sequence is: $u_n=u_1+(n-1)\cdot d$ . In S7 you discuss these sequences more in detail.

Example: An arithmetic sequence is defined by its first term $u_1 = 4$ and a common difference $d = 3$ . Find the first 5 terms of this sequence.

A geometric sequence is a sequence where every term is obtained from the previous one by multiplying it by a constant term called the ratio (or quotient). The sequence $\{2,4,6,8,10,...\}$ is not geometric, because the quotient between any two consecutive terms is not constant. E.g.: $\frac{4}{2}\neq\frac{6}{4}$ .

The general form of geometric sequences is: $u_n=u_1\cdot r^{n-1}$ . The first term and ratio are given, if we want the second term, we multiply $u_1$ by $r$ once. If we want the third term, we multiply $u_1$ by $r$ twice (so by $r^2$ ), and so on.

Example: A geometric sequence is defined by its first term $u_1 = 20$ and a common ratio $r = -0.5$ . Find the first 5 terms of this sequence, expressing any decimals or fractions clearly.

Exercises

Five sequences are explicitly defined by their general $n$ -th term formulas below for $n \ge 1$ . For each sequence, calculate the first 3 terms ( $u_1$ , $u_2$ , and $u_3$ ) by substituting the index parameters, and then determine the next 3 terms ( $u_4$ , $u_5$ , and $u_6$ ).

A sequence is explicitly modeled by a linear progression:

$u_n = 3n - 4$

A sequence is explicitly modeled by a quadratic shifting curve:

$u_n = n^2 - 4n$

A sequence is explicitly modeled by a fractional rational function:

$u_n = \frac{n + 2}{2n}$

A sequence is explicitly modeled by an alternating index rule:

$u_n = (-1)^n \cdot 2n$

A sequence is explicitly modeled by an exponential baseline power:

$u_n = 5 \cdot 2^{n-1}$

Three sequences are defined recursively below for $n \ge 1$ . For each problem, use the given first term ( $u_1$ ) and the recurrence rule to calculate the exact values for the next 5 terms ( $u_2$ , $u_3$ , $u_4$ , $u_5$ , and $u_6$ ). Simplify all fractional results completely.

An arithmetic-style shifting progression:

$u_1 = 3 \quad \text{and} \quad u_{n+1} = 2u_n - 1$

A rational fractional recurrence link:

$u_1 = 4 \quad \text{and} \quad u_{n+1} = \frac{1}{u_n} + 2$

An alternating sign quadratic operator:

$u_1 = -1 \quad \text{and} \quad u_{n+1} = (u_n)^2 - 3$

The Collatz sequence is defined by the function $f(n)$ where $u_{n+1} = f(u_n)$ :

$f(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases}$

Given the first term $u_1 = 9$ , find all subsequent terms of the sequence until it reaches the number $1$ .

Consider the arithmetic sequence defined by the ordered list parameter matrix:

$\{1, 8, 15, 22, \dots\}$

Determine the common difference ( $d$ ) of the sequence by analyzing the step offset between consecutive terms.

State the value of the first term ( $u_1$ ) and construct the general formula for the $n$ -th term ( $u_n$ ).

Calculate the value of the 10th term ( $u_{10}$ ) using an appropriate formula.

Consider the geometric sequence defined by the ordered list format:

$\{72, 36, 18, \dots\}$

Determine the common ratio ( $r$ ) of the sequence by dividing any term by its immediate preceding term.

Write out the sequence in the form $u_{n+1} = r \cdot u_n$ .

Calculate the values for the next 3 terms ( $u_4$ , $u_5$ , and $u_6$ ) of this progression, expressing any fractional values in simplified form.