1. Lesson 4-4: Arithmetic and Geometric Sequences
Advanced Math Topics

2. Arithmetic Sequence Definition: Example:
Sequence in which the difference between any term and the term before it is a constant.
i.e. you are adding or subtracting the same number each time
Example:
2, 4, 6, 8, …
You are adding 2 to each term to get the next
10, 9, 8, 7, …
You are subtracting 1 from each term to get the next

3. Common Difference Definition: Example: The value of an – an-1
2, 4, 6, 8, …
4 – 2 = 2
6 – 4 = 2
8 – 6 = 2
d= 2

4. Geometric Sequence Definition: Example:
Sequence in which the ratio of any term to the term before it is a constant.
i.e. each term is multiplied or divided by the same number
Example:
2, 4, 8, 16, …
You multiply the previous term by 2 to get the next number
81, 27, 9, 3, …
You divide each term by 3 to get the next number

5. Common Ratio The value of an divided by an-1 Example: 2, 4, 8, 16, …
4/2 = 2
8/4 = 2
16/8 = 2
So common ratio r =2

6. Arithmetic sequence 10, 7, 4, 1, … Explicit Formula Recursive Formula
an=a1 + (n-1)d
Pattern is minus 3 from previous term so d = -3
an=10 + (n-1)-3
an=10 -3n +3
an=13 – 3n
Recursive Formula
an=an-1 + d
Pattern is still minus 3 from previous term so d = -3
a1=10
an=an-1 - 3

7. Geometric Sequence 0.05, 0.25, 1.25, 6.25, … Explicit Formula
an=a1rn-1
Pattern is previous term times 5 so r = 5
Recursive Formula
an=(an-1)r
Pattern is previous term times 5 so r = 5
an=.05 (5)n-1
an =(an-1)5

8. You decide…
Is the sequence Arithmetic, geometric, or neither, justify your answer.
100, 20, 4, 0.8, …
1, 3, 6, 10, …
12, 17, 22, 27, …
0, 1, -1, 0, 1, -1, …

9. Write the recursive and explicit formulas for each sequence.
25, 22, 19, 16, …