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).