Definition:Sequence

Informal Definition
A sequence is a set of objects which is listed in a specific order, one after another.

Thus one can identify the elements of a sequence as being the first, the second, the third, ... the $n$th, and so on.

Formal Definition
A sequence is a mapping whose domain is a subset of $\N$.

It can be seen that a sequence is an instance of a family of elements indexed by $\N$.

Terms
The elements of a sequence are known as its terms.

Finite Sequence
A finite sequence is a sequence whose domain is finite.

Length of a Sequence
The length of a finite sequence is the number of terms it contains, or equivalently, the cardinality of its domain.

Sequence of n Terms
A sequence of $n$ terms is a sequence whose domain has $n$ elements.

Such a sequence is also known as an ordered n-tuple.

Null Sequence
A null sequence (or empty sequence) is one containing no terms.

Thus it is a mapping from $\varnothing$ to $S$ and therefore is null.

Infinite Sequence
An infinite sequence is a sequence whose domain is infinite.

Codomain
The codomain of a sequence can be elements of a set of any objects.

If the codomain of a sequence $f$ is $S$, then the sequence is said to be a sequence of elements of $S$, or a sequence in $S$.

Rational Sequence
A rational sequence is a (usually) infinite sequence whose codomain is the set of rational numbers $\Q$.

Real Sequence
A real sequence is a (usually) infinite sequence whose codomain is the set of real numbers $\R$.

Notation
The notation for a sequence is as follows.

If $f: A \to S$ is a sequence, then a symbol, for example "$a$", is chosen to represent elements of this sequence.

Then for each $k \in A$, $f \left({k}\right)$ is denoted $a_k$, and $f$ itself is denoted $\left \langle {a_k} \right \rangle_{k \in A}$.

Any expression can be used to denote the domain of $f$ in place of $k \in A$.

The set $A$ is usually understood to be the set $\left\{{1, 2, 3, \ldots, n}\right\}$.

If this is the case, then it is usual to write $\left \langle {a_k} \right \rangle_{k \in A}$ as $\left \langle {a_k} \right \rangle$ or even as $\left \langle {a} \right \rangle$ if brevity and simplicity improve clarity.

Sequence of Distinct Terms
A sequence of distinct terms of $S$ is an injection from a subset of $\N$ into $S$.

A sequence $\left \langle {a_k} \right \rangle_{k \in A}$ is a sequence of distinct terms iff $a_j \ne a_k$ for all $j, k \in A$ such that $j \ne k$.

Equality of Sequences
Let $f$ and $g$ be two sequences:
 * $f = \left({x_1, x_2, \ldots, x_n}\right)$
 * $g = \left({y_1, y_2, \ldots, y_m}\right)$

Then $f = g$ iff:
 * $m = n$
 * $\forall i: 1 \le i \ne n: x_i = y_i$

Notational Variants
Notation varies. Common variants for $\left \langle {a_k} \right \rangle$ are:
 * $\left({a_k}\right)$
 * $\left\{{a_k}\right\}$ (this one is not recommended though, because of the implication that the order of the terms does not matter).