9.1 Sequences and Series

Sequences

An infinite sequence is a function whose domain is the set of natural numbers
A finite sequence is a function whose domain consists of the first "n" positive integers only
In a sequence, ORDER MATTERS


Subscript can begin with 0 in some cases ()

Example 1: 

   n=      1,  2,   3,   4
=  4, 7, 10, 13

(These are the first four terms of the sequence) 

next term=now term +3 
first=4 

This sequence can be represented by: 


where  is the first term in the sequence and  is the previous term

Explicit formula: 

where  is the  term of a sequence 

*Remember: Only listing the first few terms of a sequence is not enough to define a unique sequence. To define a unique sequence, the  term must be given 

Fibonacci Sequence

The fibonacci sequence is what is called a recursive sequence. In a recursive sequence, you must be given one or more of the first few terms in order to define the sequence. 

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21



This is sequence is seen in spirals such as this one: 

*For more on how fibonacci sequences are found in nature, check out this video! : https://www.youtube.com/watch?v=ahXIMUkSXX0

Factorials

n!= 1*2*3*4....*n, where n is a natural number 

Special case: zero factorial is defined as: 0!= 1 

Example 1:

Example 2: 



   explicit formula: