1. Sequences Overview

2. Sequence
A sequence is a pattern of an ordered list of elements (numbers, figures in a picture, letters etc.)
A sequence is a function whose domain is a subset of integers (the natural #s)
A term in the sequence is an individual item / element of the list.
Examples:
3, 5, 7, 9… J, F, M, A, M, J, J,
The “…” indicates that the pattern continues.
Infinite sequences continue on forever
Finite sequences terminate

3. Term Notation
We use subscript to identify each term of the sequence starting at 1:
a1 is used to identify the 1st term in the sequence.
a12 is used to identify the 12th term in the sequence.
an is used to identify the nth term in the sequence.
Example sequence
1, 2, 4, 8, 16, 32
a1, a2, a3, a4, a5, a6, …
So a1 = 1, a2 = 2, a3 = 4, a4 = 8 etc.
Mean Same Thing

4. Types of sequences: Arithmetic Sequence
In an Arithmetic Sequence the difference between one term and the next term is a constant.
Each term is found by adding some constant value each time
always think add, add a positive or add a negative (not subtract)
The common difference is the constant number you add each time and is usually represented by the variable d
For example, Describe:
1, 4, 7, 10, 13, 16, 19, 22, 25, …
(you are adding 3 each time)
Arithmetic sequence
common difference of 3 or d=3
Made by rule (formula) an = 3n – 2.
1.5, .75, 0, -.75, -1.5, -2.25, …
(you are adding -.75 each time)
Arithmetic sequence
common difference of -.75 or d=-.75
Made by rule (formula) an = -.75n

5. Arithmetic Sequence
For each sequence, determine if it is arithmetic, and find the common difference.
-3, -6, -9, -12, …
1.1, 2.2, 3.3, 4.4, …
41, 32, 23, 14, 5, …
1, 2, 4, 8, 16, 32, …
Arithmetic, d = -3
Arithmetic, d = 1.1
Arithmetic, d = -9
Not an arithmetic sequence.

6. Types of sequences: Geometric Sequence
In a Geometric Sequence the ratio between one term and the next term is a constant
Each term is found by multiplying the pervious term by a constant.
always think multiply, multiply by an integer or by a fraction (not divide)
The common ratio is the constant number you multiply by each time and is usually represented by the variable r
For example, Describe:
2, 4, 8, 16, 32, 64, 128, …
(you are multiplying by 2)
Geometric sequence
Common ratio of 2 or r=2
27, 9, 3, 1, 1/3, 1/9, 1/27 , …
(you are multiplying by 1/3 )
Geometric sequence
Common ratio of 1/3 or r= 1/3

7. Geometric Sequence
For each sequence, determine if it is geometric, and find the common ratio.
2, 8, 32, 128, …
1, 10, 100, 1000, …
1, -1, 1, -1, …
20, 16, 12, 8, 4, …
Geometric, r = 4
Geometric, r = 10
Geometric, r = -1
Not a geometric sequence.

8. write a Sequence / Find a term
To write terms of a sequence: plug in the term number, n, as your input, and evaluate to find the term, an, as an output. (it’s just a function!)
Example 1:
Example 2:
A sequence generated by the formula
an = 6n – 4.
Generate the first 5 terms of the sequence.
a1 = 6(1) – 4
a2 = 6(2) – 4
a3 = 6(3) – 4
a4 = 6(4) – 4
a5 = 6(5) – 4
The rule is:
an = 3n + 1
Find a100
a100 = 3(100) + 1
a100 = 301
2, 8, 14, 20, 26

9. Sequences as functions
A sequence is a function whose domain is a subset of integers
domain is the set of natural numbers
remember natural numbers are the positive integers (whole numbers 1 and greater)
the term numbers are the domain (the input)
the terms of the sequence are the elements in the range (the outputs)
ALL ARITHMETIC SEQUENCES ARE LINEAR FUNCTIONS
because they have a constant rate of change (the common difference)
Geometric Sequences are other types of functions, they often are exponential (when the common ratio is positive), but can be other kinds of functions as well (when the common ratio is negative)