Describing Number and Geometric Patterns

Objectives:• Use inductive reasoning in continuing patterns• Find the next term in an Arithmetic and Geometric sequence

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

• Arrangement of geometric figures that repeat• Use inductive reasoning and make conjecture as to the next figure in a pattern

Geometric Patterns

Use inductive reasoning to find the next two figures in the pattern.

Use inductive reasoning to find the next two figures in the pattern.

Describe the figure that goes in the missing boxes.

Geometric Patterns

Describe the next three figures in the pattern below.

Numerical Sequences and Patterns

Arithmetic Sequence

Add a fixed number to the previous termFind the common difference between the previous & next term

Find the next 3 terms in the arithmetic sequence.

2, 5, 8, 11, ___, ___, ___

+3 +3 +3 +3

14

+3

17

+3

21

What is the common difference between the first and second term?

Does the same difference hold for the next two terms?

Arithmetic Sequence

17, 13, 9, 5, ___, ___, ___

What are the next 3 terms in the arithmetic sequence?

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

Term # 1 2 3 4 n

Term 17 13 9 5

A(1) = 17A(2) = 17 + (-4)A(3) = 17 + (-4) + (-4)A(4) = 17 + (-4) + (-4) + (-4)

Relate

Formula A(n) = 17 + (n – 1)(-4)

Arithmetic Sequence Rule

nth term

firstterm

termnumber

Commondifference

Find the first, fifth, and tenth term of the sequence: A(n) = 2 + (n - 1)(3)

A(n) = 2 + (n - 1)(3)

First Term

A(1) = 2 + (1 - 1)(3)

= 2 + (0)(3)

= 2

A(n) = 2 + (n - 1)(3)

Fifth Term

A(5) = 2 + (5 - 1)(3)

= 2 + (4)(3)

= 14

A(n) = 2 + (n - 1)(3)

Tenth Term

A(10) = 2 + (10 - 1)(3)

= 2 + (9)(3)

= 29

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)

Real-world and Arithmetic Sequence

What is the function rule?

.32 + 23 .32+.23+.23 .32+.23+.23+.23

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) = 2.39The cost is $2.39.

3, 12, 48, 192, ___, _____, ______12,288

Numerical Sequences and Patterns

Geometric Sequence

• Multiply by a fixed number to the previous term• The fixed number is the common ratio

Find the common ratio and the next 3 terms in the sequence.

x 4 x 4 x 4 x 4

768

x 4

3072

x 4What is the common RATIO between the first and second term?

Does the same RATIO hold for the next two terms?

Geometric Sequence

80, 20, 5, , ___, ___

What are the next 2 terms in the geometric sequence?

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

Term # 1 2 3 4 n

Term 80 20 5

A(1) = 80A(2) = 80 · (¼)A(3) = 80 · (¼) · (¼)

A(4) = 80 · (¼) · (¼) · (¼)

Relate

Formula A(n) = 80 · (¼)n-1

4

516

5

64

5

4

1

4

5

Geometric Sequence Rule

nth term

firstterm

commonratio

Term number

Find the first, fifth, and tenth term of the sequence: A(n) = 2 · 3n - 1

A(n) = 2· 3n - 1

First Term

A(n) = 2 · 3n - 1

Fifth Term

A(n) = 2· 3n - 1

Tenth Term

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

Write a Function Rule

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.

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