1. Arithmetic and geometric sequences
Lesson 2
Arithmetic and geometric sequences

2. 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
1.5, .75, 0, -.75, -1.5, -2.25, …
(you are adding -.75 each time)
Arithmetic sequence
common difference of -.75 or d=-.75

3. 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.

4. 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

5. 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.

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

7. write a Sequence / Find a term
To write terms of a sequence or find a term: plug in the term number, n, as input, and evaluate to find the term, an, as an output. (it’s just a function!)
Example 3:
A sequence generated by the formula
an = 5 + 2(n –1).
Generate the first 5 terms of the sequence.
a1 = 5 + 2((1) – 1) = 5
a2 = 5 + 2((2) – 1) = 7
a3 = 5 + 2((3) – 1) = 9
a4 = 5 + 2((4) – 1) = 11
a5 = 5 + 2((5) – 1) = 13
Example 4:
A sequence generated by the formula
xn = 2(n –1)
List the first 5 terms.
x1 = 2((1) – 1)
x2 = 2((2) – 1)
x3 = 2((3) – 1)
x4 = 2((4) – 1)
x5 = 2((5) – 1)
= 2(0) = 1
= 2(1) = 2
= 2(2) = 4
= 2(3) = 8
= 2(4) = 16
5, 7, 9, 11, 13
1, 2, 4, 8, 16

8. Sequences as functions
Remember a sequence is a function 
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, but can be other kinds of functions as well