9.1 Sequences and Series

9.1 Sequences and Series

Sequences:

A sequence is a function whose domain is the set of positive integers.

In an infinite sequence, the domain of the function is the set of positive integers
    The function values are as follows:

         

In a finite sequence, the domain of the function is defined as the first n positive integers

To find terms in a sequence you must plug the set of positive integers wherever n may be present and solve for the equation

For example:
     Find the first 3 terms of the sequences given by

 

You now know from the equation that the values imputed are: 

Now just plug in for values of n:

Recursive Sequences:

A recursive sequence is a sequence in which all terms are defined using previous terms.
To find a recursive sequence you must be given one or more of the first few terms.
For example:
     Find the first 3 terms of the sequence defined recursively

Since you are given what is necessary, you may now plug values in and solve the sequence

To continue on and solve forand   you must plug in the previously found answer in the sequence 

Factorials:

Factorials are a type of term that is important to solving sequences

They are as follow:

It is important that you know 

Explicit Sequences:

In explicit sequences, any term may be found without having to know every term before. In the sequence  any integer may be replaced for n to find its value. If you are trying to find  plug 100 in for n and solve: