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 isare:

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,… ).