1. Introduction to Sequences
Dr. Shildneck

2. Sequences a1, a2, a3, a4, a5, …, an, … a1, a2, a3, a4, a5, …, an
An Infinite sequences (an) is a function whose domain is the set of positive integers.
The function values, called the terms of the sequence, are represented by
a1, a2, a3, a4, a5, …, an, …
Sequences whose domain consist of only the first n positive integers are called a finite sequence.
a1, a2, a3, a4, a5, …, an

3. Sequences as Functions
We can (and have) looked at sequences as functions. When doing so, we consider the following.
“Sequence Functions” are discrete. This means that the graphs produce a pattern of points, not connected lines or curves.
The domain of the function is the position number (n) of the term.
The range of the function is the value of the actual term that is in that position. (an)

4. Rules for Sequences
There are two different types of rules that we can write for sequences.
First, an explicit (or closed) rule is written as a function with independent variable n and dependent variable an.
The second form is called a recursive rule. Recursive rules require 2 (or more) parts including an initial term and a rule that explains how to get each consecutive term based on the previous term(s).

5. Rules for Sequences
While most sequences can easily be written in either form, there are advantages and disadvantages of both.
Explicit Rules allow us to quickly determine a term value by simply plugging in the position value that we need to know.
Very complex patterns are often easier to express by recursive rules because of their complex formulas or patterns.

6. Example Using the Recursive Rule
Consider the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Write a recursive rule for the pattern.
Find a1
Find a4
Find a23

7. The Fibonacci Sequence
The previous sequence is the famed Fibonacci sequence (from the DaVinci code).
As an example of how difficult the explicit formulas can be, the following equation will work for this sequence for n<1475.

8. Examples
Find the first 5 terms defined by the following definition.
a1 = 7
an = an-1 + 3

9. Examples
Find the first 5 terms defined by the following definition.
an = −1 𝑛 3 𝑛 −1

10. Examples
Write the recursive and explicit formula for the following sequence.
5, 10, 15, 20, 25, 30, 35, 40, …

11. Examples
Write an explicit formula for the nth term of the following sequence.
5 3 , 6 6 , 7 9 , 8 12 , , …

12. Examples
Write an explicit formula for the nth term of the following sequence.
1 2 ,− 1 4 , 1 8 ,− 1 16 , , …

13. Examples
Write an explicit formula for the nth term of the following sequence.
1, 4, 9, 16 ,25, 36, 49, …

14. ASSIGNMENT
ASSIGNMENT 1 – INTRO TO SEQUENCES WS