1. Describing Number and Geometric Patterns
Objectives:
Use inductive reasoning in continuing patterns
Find the next term in an Arithmetic and Geometric sequence
Vocabulary
Inductive reasoning:
make conclusions based on patterns you observe
Conjecture:
conclusion reached by inductive reasoning based on evidence
Geometric Pattern:
arrangement of geometric figures that repeat
Arithmetic Sequence
Formed by adding a fixed number to a previous term
Geometric Sequence
Formed by multiplying by a fixed number to a previous term

2. Geometric Patterns
Arrangement of geometric figures that repeat
Use inductive reasoning and make conjecture as to the next figure in a pattern
Use inductive reasoning to find the next two figures in the pattern.
Use inductive reasoning to find the next two figures in the pattern.

3. Do Now Geometric Patterns
Describe the figure that goes in the missing boxes.
Describe the next three figures in the pattern below.

4. Numerical Sequences and Patterns
Arithmetic Sequence
Add a fixed number to the previous term
Find the common difference between the previous & next term
Example
Find the next 3 terms in the arithmetic sequence.
2, 5, 8, 11, ___, ___, ___ 14 17 21 +3 +3 +3 +3 +3 +3
What is the common difference between the first and second term?
Does the same difference hold for the next two terms?

5. What are the next 3 terms in the arithmetic sequence?
17, 13, 9, 5, ___, ___, ___ 1 -3 -7
An arithmetic sequence can be modeled using a function rule.
What is the common difference of the terms in the preceding problem? -4
Let n = the term number
Let A(n) = the value of the nth term
in the sequence
A(1) = 17
A(2) = 17 + (-4)
A(3) = 17 + (-4) + (-4)
A(4) = 17 + (-4) + (-4) + (-4)
Term # 1 2 3 4 n
Term 17 13 9 5
Relate
Formula A(n) = 17 + (n – 1)(-4)

6. Arithmetic Sequence Rule
A(n) = a + (n - 1) d
Common
difference
nth
term
first
term
term
number
Find the first, fifth, and tenth term of the sequence:
A(n) = 2 + (n - 1)(3)
First Term
Fifth Term
Tenth Term
A(n) = 2 + (n - 1)(3)
A(n) = 2 + (n - 1)(3)
A(n) = 2 + (n - 1)(3)
A(1) = 2 + (1 - 1)(3)
A(5) = 2 + (5 - 1)(3)
A(10) = 2 + (10 - 1)(3)
= 2 + (0)(3)
= 2 + (4)(3)
= 2 + (9)(3)
= 2
= 14
= 29

7. Real-world and Arithmetic Sequence
In 1995, first class postage rates were raised to 32 cents for the first ounce and 23 cents for each additional ounce. Write a function rule to model the situation.
Weight (oz)
A(1)
A(2)
A(3) n
Postage (cents)
What is the function rule?
A(n) = (n – 1)(.23)
What is the cost to mail a 10 ounce letter?
A(10) = (10 – 1)(.23)
= (9)(.23)
= 2.39
The cost is $2.39.

8. Numerical Sequences and Patterns
Geometric Sequence
Multiply by a fixed number to the previous term
The fixed number is the common ratio
Example
Find the common ratio and the next 3 terms in the sequence.
3, 12, 48, 192, ___, _____, ______
768
3072
12,288
x 4
x 4
x 4
x 4
x 4
x 4
Does the same RATIO hold for the next two terms?
What is the common RATIO between the first and second term?

9. What are the next 2 terms in the geometric sequence?
80, 20, 5, , ___, ___
An geometric sequence can be modeled using a function rule.
What is the common ratio of the terms in the preceding problem?
Let n = the term number
Let A(n) = the value of the nth term
in the sequence
A(1) = 80
A(2) = 80 · (¼)
A(3) = 80 · (¼) · (¼)
A(4) = 80 · (¼) · (¼) · (¼)
Term # 1 2 3 4 n
Term 80 20 5
Relate
Formula A(n) = 80 · (¼)n-1

10. Geometric Sequence Rule
A(n) = a r
Term
number
nth
term
first
term
common
ratio
Find the first, fifth, and tenth term of the sequence:
A(n) = 2 · 3n - 1
First Term
Fifth Term
Tenth Term
A(n) = 2· 3n - 1
A(n) = 2 · 3n - 1
A(n) = 2· 3n - 1
A(1) = 2·
A(5) = 2 ·
A(10) = 2·
A(1) = 2
A(5) = 162
A(10) = 39,366

11. Real-world and Geometric Sequence
You drop a rubber ball from a height of 100 cm and it bounces back to lower and lower heights. Each curved path has 80% of the height of the previous path. Write a function rule to model the problem.
Write a Function Rule
A(n) = a· r n - 1
A(n) = 100 · .8 n - 1
What height will the ball reach at the top of the 5th path?
A(n) = 100 · .8 n - 1
A(5) = 100 ·
A(5) = cm