# What is a sequence?

Oct 19, 2014

A sequence is simply an ordered list like:

$\left\{1 , 2 , 3 , 4 , \ldots\right\}$

and

$\left\{1 , \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , \ldots\right\}$.

I hope that this was helpful.

Sep 7, 2017

A sequence is an infinite set of points which are ordered and a mapping can be associated with with that set and the set of natural numbers.

The general notion of a sequence is that it is an infinite set with every element associated with a natural number even though all infinite sets may not be sequences.

A sequence may be represented by, $\left\{{x}_{n}\right\}$ where ${x}_{n}$ is the $n$th element related to the a corresponding natural number.

Thus if ${x}_{n} = \frac{1}{n} ^ 2$, the sequence may be given as,

$\left\{1 , \frac{1}{4} , \frac{1}{9} , \frac{1}{16} , \ldots .\right\}$

For ${x}_{n} = {n}^{3}$ we shall have,

$\left\{1 , 8 , 27 , 64 , \ldots .\right\}$

Now for ${x}_{n} = n$ we can have,

$\left\{1 , 2 , 3 , \ldots . .\right\}$

This is indeed the set of naturals.

However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. They don't fit right with the definition I gave.

Let's for example take a Fibonacci sequence.

$1 , 1 , 2 , 3 , 5 , 8 , 13 , \ldots .$

This sequence is made by adding the previous two numbers on the list to form the next one and so on.

There can be arithmetic sequences, like

$2 , 8 , 14 , 20 , \ldots .$ which has first term $2$ and common difference $6$.

I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs.

Sep 7, 2017

See explanation...

#### Explanation:

A well ordered set is a set of objects together with an ordering relation $<$ between any two elements such that any non-empty subset has a least element. The simplest example of a well ordered infinite set is the natural numbers with their usual ordering.

A sequence is a function from a well ordered set to another set. The domain is known as the index, and the terms of the sequence are often written with the index as a subscript, e.g. ${a}_{n}$.

Normally the domain is either finite, being the set of all natural numbers up to a given natural number, or is infinite, being the set of all natural numbers. It is also possible to have sequences larger than the set of natural numbers, indexed by well ordered sets larger than the natural numbers.

More simply, a finite sequence is a list of elements:

${a}_{1} , {a}_{2} , {a}_{3} , \ldots , {a}_{n}$

and most infinite sequences you will encounter are written:

${a}_{1} , {a}_{2} , {a}_{3} , \ldots$

where the ellipsis signifies that the list continues with no last element.

Typical examples you will meet at precalculus level are arithmetic sequences and geometric sequences, but these are only very particular kinds of sequence. There are many other useful types and examples of sequence.

For a use of transfinite sequences indexed by well-ordered sets larger than $\mathbb{N}$, see https://socratic.org/s/aSFj2GUc