Lesson 4-4: Arithmetic and Lesson 4-4: Arithmetic and Geometric SequencesGeometric Sequences

Advanced Math TopicsAdvanced Math Topics

Arithmetic SequenceArithmetic Sequence

Definition:Definition: Sequence in which the Sequence in which the

difference between difference between any term and the term any term and the term before it is a constant.before it is a constant.

i.e. you are adding or i.e. you are adding or subtracting the same subtracting the same number each timenumber each time

Example:Example: 2, 4, 6, 8, …2, 4, 6, 8, … You are adding 2 to You are adding 2 to

each term to get the each term to get the nextnext

10, 9, 8, 7, …10, 9, 8, 7, … You are subtracting 1 You are subtracting 1

from each term to get from each term to get the nextthe next

Common DifferenceCommon Difference

Definition:Definition: The value of aThe value of ann – a – an-1n-1

Example:Example: 2, 4, 6, 8, …2, 4, 6, 8, …

4 – 2 = 24 – 2 = 2 6 – 4 = 26 – 4 = 2 8 – 6 = 28 – 6 = 2

d= 2d= 2

Geometric SequenceGeometric Sequence

Definition:Definition: Sequence in which the Sequence in which the

ratio of any term to the ratio of any term to the term before it is a term before it is a constant.constant.

i.e. each term is i.e. each term is multiplied or divided by multiplied or divided by the same numberthe same number

Example:Example: 2, 4, 8, 16, …2, 4, 8, 16, … You multiply the You multiply the

previous term by 2 to previous term by 2 to get the next numberget the next number

81, 27, 9, 3, …81, 27, 9, 3, … You divide each term You divide each term

by 3 to get the next by 3 to get the next numbernumber

Common RatioCommon Ratio

The value of aThe value of ann

divided by adivided by an-1n-1

Example:Example: 2, 4, 8, 16, …2, 4, 8, 16, … 4/2 = 24/2 = 2 8/4 = 28/4 = 2 16/8 = 216/8 = 2

So common ratio r =2So common ratio r =21n

n

a

a

Arithmetic sequence 10, 7, 4, 1, …Arithmetic sequence 10, 7, 4, 1, …

Explicit FormulaExplicit Formula aann=a=a11 + (n-1)d + (n-1)d

Pattern is minus 3 Pattern is minus 3 from previous term so from previous term so d = -3d = -3

aann=a=a11 + (n-1)d + (n-1)d

aann=10 + (n-1)-3=10 + (n-1)-3

aann=10 -3n +3=10 -3n +3

aann=13 – 3n=13 – 3n

Recursive FormulaRecursive Formula aann=a=an-1n-1 + d + d

Pattern is still minus 3 Pattern is still minus 3 from previous term so from previous term so d = -3d = -3

aann=a=an-1n-1 + d + d

aa11=10=10

aann=a=an-1n-1 - 3 - 3

Geometric Sequence 0.05, 0.25, Geometric Sequence 0.05, 0.25, 1.25, 6.25, …1.25, 6.25, …

Explicit FormulaExplicit Formula aann=a=a11rrn-1n-1

Pattern is previous Pattern is previous term times 5 so r = 5term times 5 so r = 5

aann=a=a11rrn-1n-1

Recursive FormulaRecursive Formula aann=(a=(an-1n-1)r)r

Pattern is previous Pattern is previous term times 5 so r = 5term times 5 so r = 5

aann=(a=(an-1n-1)r)raann=.05 (5)n-1=.05 (5)n-1

aann =(a =(an-1n-1)5)5

You decide…You decide…

Is the sequence Arithmetic, geometric, or Is the sequence Arithmetic, geometric, or neither, justify your answer.neither, justify your answer. 100, 20, 4, 0.8, …100, 20, 4, 0.8, …

1, 3, 6, 10, …1, 3, 6, 10, …

12, 17, 22, 27, …12, 17, 22, 27, …

0, 1, -1, 0, 1, -1, …0, 1, -1, 0, 1, -1, …

Write the Write the recursive and explicitrecursive and explicit formulas for each sequence.formulas for each sequence.

25, 22, 19, 16, …25, 22, 19, 16, …

,...81

2,

27

2,

9

2,

3

2