# Sequences

Note the sequence of years in which the Olympics were held, starting in 1996:

(1996, 2000, 2004, 2008, 2012, 2016… )

The parentheses suggest that we are working with a set of numbers placed in a certain order. These elements are called sequence terms. Usually each term of a sequence is represented by any letter, usually The, accompanied by an index that gives its position or order. For example, following (1996, 2000, 2004, 2008,… ), we have:

• first term = = 1996;
• second term = = 2000;
• third term = = 2004;
• fourth term = = 2008;
• … (and so on).

The nth term can represent any term in the sequence. For example, if n = 50we have and we are referring to 50th term of the sequence.

## Sequence Definition

Mathematically, sequence is called any function f whose domain is .

Example defined by f (n) = 2n

Replacing Yourself no by natural numbers 1, 2, 3,… we have: Therefore, the sequence can be written as (2, 4, 6,…, 2n,…).

Note that there is a training law of the terms of a sequence. From now on, we will study two different ways of defining a sequence: by the general term and by recurrence.

### Sequence defined by the general term

Each term is calculated as a function of your position no in sequence.

Example

The first three terms of the sequence whose general term is are: So the sequence that has as its general term , é .

### Sequence defined by recurrence

Each term in the sequence is calculated against the previous term.

Example

In the sequence defined by on what , each term except the first is the same as the previous one added to 3. Therefore, the sequence can be written as (4, 7, 10, 13,… ).

Next: Arithmetic Progression