## FANDOM

1,019 Pages

A sequence is an ordered set of objects. A sequence that goes on forever is called an infinite sequence, whereas one that does not is called a finite sequence. The sum of a sequence is called a series.

Sequences are usually denoted as

$\{a_k\}_m^n$

with $k$ being the term number and $m,n$ being the bounds of the series.

An example would be

$\{a_k\}_{m=1}^\infty=3^k=3,9,27,81,243,\ldots$

Sequences described with the previous terms are called recursive sequences. For instance, the Fibonacci numbers can be described as

$a_0=0,a_1=1$
$\{a_k\}_{m=0}^\infty=a_{k-1}+a_{k-2}$

Two common types are arithmetic and geometric sequences. Arithmetic sequences have a given difference between each term. For example,

$\{a_k\}_{m=0}^\infty=2k+1=1,3,5,7,\ldots$

Geometric sequences take the form

$\{a_k\}_{m=0}^\infty=ar^k$

Where $r$ is a common ratio.

## Sets vs Sequences

Unlike a set , sequences allow repeats and the order matters.

## Formal definition

Let $S$ be a set
Let $\N$ be the set of natural numbers.
Then, a mapping $a:\N\to S$ is called a sequence of elements of $S$ . The image of an element of under $a$ (that is $a(i)$) is denoted as $S_i$ , where $i\in\N$ .

An equivalent definition is an indexed family indexed by the natural numbers.

While the formal definition of a sequence is a treats a sequence as a function, in practice they are treated somewhat like a set with "order".