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

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

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

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

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

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

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

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, …

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