Describing Number and Geometric Patterns

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
Arithmetic Sequence
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)
.32 23
.32.23.23
.32.23.23.23
What is the function rule?
A(n) .32 (n 1)(.23)
What is the cost to mail a 10 ounce letter?
A(10) .32 (10 1)(.23) .32
(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, ___, _____, ______
12,288
768
3072
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
Geometric Sequence
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
n-1
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 31 - 1
A(5) 2 35 - 1
A(10) 2 310 - 1
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 .8 5 - 1
A(5) 40.96 cm