Sequences in mathematics
A sequence is a function that assigns to each natural number n a real number an. It consists of an ordered list of countable elements a1, …, an, known as the terms of the sequence. $$ \{ a_n \} = a_1, a_2, a_3, \dots, a_n $$ In this context, n is called the index of the sequence, while an is referred to as the general term of the sequence.
A sequence is formally defined as a function f whose domain is the set of natural numbers (N), and whose range is either the real numbers (R) or the natural numbers (N).
$$ a: N \rightarrow R $$
A sequence may contain either a finite or an infinite number of terms.
$$ \{ a_n \} = a_1, a_2, a_3, \dots, a_n $$
The term an is called the n-th term or general term of the sequence. It’s pronounced “a sub n.”
Each term in the sequence, such as a1, a2, a3, …, is labeled with a subscript written slightly lower and to the right of the letter, indicating the position of the element within the list.
The term a1 is the first term of the sequence, a2 is the second, and so on.
Note. The subscripts in a sequence increase in ascending order. A sequence usually begins at n = 1 or n = 0, although it’s entirely possible to define a starting index greater than one.
An Example of a Sequence
Example 1
Consider the sequence 2n:
$$ a_n = 2n $$
The first few terms of the sequence are:
$$ a_1 = 2 \cdot 1 = 2 \\ a_2 = 2 \cdot 2 = 4 \\ a_3 = 2 \cdot 3 = 6 \\ a_4 = 2 \cdot 4 = 8 \\ \vdots $$
So, the sequence can be written as:
$$ \{ 2n \} = 2, \; 4, \; 6, \; 8, \; \dots $$
Example 2
Consider the sequence n2:
$$ a_n = n^2 $$
The first few terms of this sequence are:
$$ a_1 = 1^2 = 1 \\ a_2 = 2^2 = 4 \\ a_3 = 3^2 = 9 \\ a_4 = 4^2 = 16 \\ \vdots $$
Hence, the sequence is:
$$ \{ n^2 \} = 1, \; 4, \; 9, \; 16, \; \dots $$
Example 3
Consider the sequence 1/n:
$$ a_n = \frac{1}{n} $$
The first few terms are:
$$ a_1 = \frac{1}{1} = 1 \\ a_2 = \frac{1}{2} \\ a_3 = \frac{1}{3} \\ a_4 = \frac{1}{4} \\ \vdots $$
Thus, the sequence can be written as:
$$ \left\{ \frac{1}{n} \right\} = 1, \; \frac{1}{2}, \; \frac{1}{3}, \; \frac{1}{4}, \; \dots $$
Ways to Represent Sequences
There are several ways to represent a sequence:
1] Enumeration
A sequence can be represented by listing its first few terms, typically four or five, followed by an ellipsis:
$$ 2, \; 4, \; 9, \; 16, \; \dots $$
The position of each term in the list indicates its index in the sequence.
For the sequence above, 2 is the first term, 4 is the second, and so forth.
Note. Since there are infinitely many natural numbers, it’s impossible to list every term in an infinite sequence. However, the first few terms are usually sufficient to identify the underlying pattern and deduce the general term.
Enumeration is simple and intuitive, but it can sometimes lead to ambiguity.
For example, two different sequences might share the same first four or five terms but diverge afterwards. Which sequence should you choose?
Moreover, the initial terms might not always make the general term of the sequence obvious.
To avoid uncertainty, it’s preferable, whenever possible, to express a sequence using an analytical formula.
2] Analytical Representation
In an analytical representation, a sequence is defined by its general term:
$$ a_n = 2n - 1 $$
This leaves no room for ambiguity about how the sequence behaves.
Note. Unfortunately, it’s not always straightforward to express a sequence in terms of its general term.
3] Recursive Representation
Another way to define a sequence is through recursion.
In a recursive representation, each term is defined in terms of the preceding one using mathematical recursion.
When using a recursive definition, you must specify:
- the initial term of the sequence (a0), and
- the relation linking an to the preceding term an-1.
This method of representation is particularly useful in the study of complex systems.
Example. Consider the following recursively defined sequence: $$ \begin{cases} a_0 = 1 \\ \\ a_n = a_{n-1} + 2 \cdot n \end{cases} $$ The first few terms of this sequence are: $$ a_0 = 1 $$ $$ a_1 = a_0 + 2 \cdot 1 = 1 + 2 \cdot 1 = 3 $$ $$ a_2 = a_1 + 2 \cdot 2 = 3 + 2 \cdot 2 = 7 $$ $$ a_3 = a_2 + 2 \cdot 3 = 7 + 2 \cdot 3 = 13 $$ Therefore, the initial terms of the sequence are: $$ 1, \; 3, \; 7, \; 13, \; \dots $$
Sequences and the Cartesian Product
A sequence has its domain in the natural numbers N (or a subset of N) and its codomain S in either the natural or real numbers. Thus, a sequence can be viewed as a special case of the Cartesian product N×S. $$ (n, a_n) $$
The first element (n) of the pair represents the index indicating the term’s position within the sequence.
The second element (an) is the value of the n-th term in the sequence.
Example
Consider the sequence {an} = 2n:
$$ a_1 = 2 \\ a_2 = 4 \\ a_3 = 6 \\ a_4 = 8 $$
The corresponding Cartesian product (n, an) is:
$$ (1, 2) \\ (2, 4) \\ (3, 6) \\ (4, 8) $$
Plotting these points on a Cartesian diagram provides a clear visual representation:
The Difference Between a Set and a Sequence
In a sequence, the order of elements is crucial, whereas in a set, the order does not matter.
Example
Sets A and B are equal because, in a set, the order of elements is irrelevant:
$$ A = \{ 1, 2, 3, 4 \} $$
$$ B = \{ 3, 1, 4, 2 \} $$
By contrast, the sequences {a} and {b} are distinct:
$$ \{a\} = \{ 1, 2, 3, 4 \} $$
$$ \{b\} = \{ 3, 1, 4, 2 \} $$
What Are Strings?
A sequence that contains a finite number of elements is called a string.
The number of elements it contains is called the length of the string.
Example
If n = 4, the sequence {an} = 2n is considered a string:
$$ a_1, a_2, a_3, a_4 = 2, 4, 6, 8 $$
The length of this string is 4.
And so on.
