Sequence

Definition of Sequence

A Sequence is a set of numbers which are written in order.

• In mathematics, is a string of objects, like numbers, that follow particular pattern
• The individual elements in a sequence are called terms
• Some of the simplest sequences can be found in multiplication tables

Types of Sequence

• Arithmetic Sequence
• Geometric Sequence
• Harmonic Sequence
• Fibonacci Sequence

Infinite or Finite of Sequence     

When the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence.

Examples

In Order

When we say the terms are “in order”, we are free to define what order that is! They could go forwards, backwards or they could alternate or any type of order we want!

Like a Set

A Sequence is like a Set, except :

• The terms are in order (with sets the order does not matter)
• The same value can appear many time (only once in sets)

Example

Notation of Sequence

Sequences also use the same notation a sets :

Notes : The curly brackets { } are sometimes called “set brackets” or “braces”.

A Rule of Sequence

A sequence usually has a Rule, which is a way to find the value of each term.

Example

As a Formula

So, What Can A Rule For {3, 5, 7, 9, ...} Be?

That nearly worked, but it is too low by 1 every time, so let us try changing it to :

That Works!

Now we can calculate.

Example

Many Rules

But mathematics is so powerful, we can find more than one Rule that works for any sequence.

Example

But can we find another rule?

A completely different sequence!

So it is best to say “A Rule” rather than “The Rule” (unless we know it is the right Rule).

Notation

To make it easier to use rules, we often use this special style :

Example

Example

Calculations :

Here are the examples :

Example 1

Example 2

This is the sequence of square numbers.

Example 3

Example 4

Example 5

References :

• http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-apgp-2009-1.pdf
• http://www.mathsisfun.com/algebra/sequences-series.html