Sequence

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, and $a_k$ being the term, usually found with a function $a_k = f(k)$ or rule describing them.

An example would be

$a_k = 3^k$

$(a_k)_{m=1}^\infty=(3,9,27,81,243,\cdots)$

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

$a_0=0,a_1=1, a_k = a_{k-1}+a_{k-2}$

$(a_k)_{m=0}^{\infty}=(0,1,1,2,3,5,8,\cdots)$

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

$a_k = 2k+1$

$(a_k)_{m=0}^{\infty}=(1,3,5,7,\cdots)$

Geometric sequences take the form

$a_k = ar^k$

$(a_k)_{m=0}^{\infty}=(a,ar,ar^2,ar^3,\cdots)$

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 $\mathbb{N}$ 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".