INTRODUCTION TO RECURSION AND SEQUENCES
DEFINITION sequence; notation for sequences
A sequence is an ordered list of numbers.
Each number in the sequence is called a term.
The [beautiful math coming... please be patient] $\,n^{\text{th}}\,$ term can be denoted as:
  • [beautiful math coming... please be patient] $\,u_n\,$ (subscript notation); or
  • $\,u(n)\,$ (function notation)
EXAMPLES:
The first five terms of the sequence defined by [beautiful math coming... please be patient] $\,u(n) = n^2\,$ are:
$u(1) = 1^2 = 1$
$u(2) = 2^2 = 4$
$u(3) = 3^2 = 9$
$u(4) = 4^2 = 16$
$u(5) = 5^2 = 25$
The [beautiful math coming... please be patient] $\,27^{\text{th}}\,$ term of the sequence defined by $\,u_n = n + 3\,$ is:
$u_{27} = 27 + 3 = 30$
DEFINITION recursion
Recursion is a process in which each step of a pattern is dependent on the step or steps that came before it.
DEFINITION recursive formula
A recursive formula must specify:
  • one (or more) starting terms
  • a recursive rule that defines the $\,n^{\text{th}}\,$ term in relation to previous term(s)
EXAMPLE:
The recursive rule
$\,u_1 = 2\,$;
[beautiful math coming... please be patient] $\,u_n = u_{n-1} + 3\,$ for $\,n\ge 2\,$
generates the sequence   $\,2\,$, $\,5\,$, $\,8\,$, $\,11\,$, $\,14\,$, $\,\ldots\,$
Thought process:
Start with the number $\,2\,$.$\,u_1=2\,$ tells you this;
$\,u_1\,$ represents the first term in the sequence $\,u\,$
To find any other term,
take the previous term and add $\,3\,$.
$\,u_n = u_{n-1} + 3\,$ for $\,n\ge 2\,$ tells you this.
For example, suppose $\,n = 2\,$, so you're looking at: $\,u_2 = u_{2-1} + 3 = u_1 + 3\,$
How do you get the second term, $\,u_2\,$?
Answer: take the first term, $\,u_1\,$, and add $\,3\,$ to it.
EXAMPLE:
The recursive rule
$w_1 = 1\,$,   $\,w_2 = 1\,$;
[beautiful math coming... please be patient] $w_n = w_{n-1} + w_{n-2}\,$ for $\,n\ge 3$
generates the sequence   $\,1\,$, $\,1\,$, $\,2\,$, $\,3\,$, $\,5\,$, $\,8\,$, $\,13\,$, $\,\ldots$
Thought process:
Start with the numbers $\,1\,$ and $\,1\,$.
To find any other term, take the previous two terms and add them together.
DEFINING A SEQUENCE BOTH RECURSIVELY AND NONRECURSIVELY

Some sequences can be defined both recursively and non-recursively.
For example, the sequence   $\,3\,$, $\,5\,$, $\,7\,$, $\,9\,$, $\,11\,$, $\,\ldots$
can be defined in either of the following ways:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Arithmetic and Geometric Sequences


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7
8 9 10 11 12 13 14
AVAILABLE MASTERED IN PROGRESS

(MAX is 14; there are 14 different problem types.)