Introduction to Sequences

A sequence is an ordered list of terms or elements. And we say also a sequence can be thought of as a list of numbers written in a definite order . Each term in a sequence is identified by its location in the sequence.

 is the first term.

is the second term.

is the third term.

is the fourth term.

 and so on...

EXAMPLES 1: Some sequences can be defined by giving a formula for the nth term. In the following examples we give three descriptions of the sequence: one by using the preceding notation, another by using the defining formula, and a third by writing out the terms of the sequence. Notice that doesn’t have to start at 1.

A sequence can be finite or infinite.

 A finite sequence is a function with domain 1, 2, 3, ..., n.

An infinite sequence  is a function with domain 1, 2, 3, 4, ...

A sequence is often expressed as a rule or formula.

EXAMPLE 2: Find the first four terms of sequence   

EXAMPLE 3: Find the first four terms of sequence   

A sequence can also be expressed as a recursive formula, this means each term in the sequence is based on previous terms, not just n. For example:  

Another example:

There are different types of sequences.

An arithmetic sequence is a sequence that has the pattern of adding the same value to determine consecutive terms. 

We say arithmetic sequences have a common difference, , example:

A geometric sequence is a sequence that has the pattern of multiplying by a constant to determine consecutive Terms.

 We say geometric sequences have a common ratio,   , For example: