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Lesson 7-1. Geometric Mean. Sequences. Arithmetic Sequence:. Is a pattern of numbers where any term (number in the sequence) is determined by adding or subtracting the previous term by a constant called the common difference . 17 20 23. Example:.

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lesson 7 1
Lesson 7-1

Geometric Mean

Lesson 7-1: Geometric Mean

sequences
Sequences

Arithmetic Sequence:

Is a pattern of numbers where any term (number in the sequence) is determined by adding or subtracting the previous term by a constant called the common difference.

17 20 23

Example:

2, 5, 8, 11, 14, ____, ____, ____

Common difference = 3

Geometric Sequence:

Is a pattern of numbers where any term (number in the sequence) is determined by multiplying the previous term by a common factor.

Example:

2, 6, 18, 54, 162, _____, _____, ____

486 1458 4374

Common Factor = 3

Lesson 7-1: Geometric Mean

examples
Examples

1. Starting with the number 1 and using a factor of 4, create 5 terms of a geometric sequence.

1 , 4 , 16 , 64 , 256

2. Starting with the number 2 and using a factor of 5, create 5 terms of a geometric sequence.

2 , 10 , 50 , 250 , 1250

3. Starting with the number 5 and using a factor of 3, create 5 terms of a geometric sequence.

5 , 15 , 45 , 135 , 405

4. In the geometric sequence 2, ____, 72, 432, .Find the missing term.

12

12

5. In the geometric sequence 6, ____, 24,... Find the missing term.

Lesson 7-1: Geometric Mean

geometric mean
Geometric Mean

A term between two terms of a geometric sequence is the geometric mean of the two terms.

Example:

In the geometric sequence 4, 20, 100, ….(with a factor of 5), 20 is the geometric mean of 4 and 100.

Try It:

Find the geometric mean of 3 and 300.

3 , ___ , 300

30

Lesson 7-1: Geometric Mean

geometric mean fact
Geometric Mean : Fact

Consecutive terms of a geometric sequence are proportional.

Example:

Consider the geometric sequence with a common factor 10.

4 , 40 , 400

(4)(400) = (40)(40)

cross-products are equal

1600 = 1600

Lesson 7-1: Geometric Mean

therefore
Therefore ………..

To find the geometric mean between 7 and 28 ...

label the missing term x

7 , ___ , 28

write a proportion

cross multiply

solve

Lesson 7-1: Geometric Mean

the geometric mean between two numbers a and b is the positive number x where therefore x
The geometric mean between two numbers a and b is the positive number x where . Therefore x = .

Try It:

Find the geometric mean of . . .

Answer = 20

1. 10 and 40

2. 1 and 36

Answer = 6

3. 10 and 20

Answer = 14.14

4. 5 and 6

Answer = 5.48

Answer = 9.94

5. 8.1 and 12.2

Lesson 7-1: Geometric Mean

how does this relate to geometry
How does this relate to geometry?

Lesson 7-1: Geometric Mean

slide9

The " W " Pattern

Lesson 7-1: Geometric Mean

the geometric means
The Geometric Means

Recall the three geometric means that you

discovered from your Sketchpad activity.

BUT FIRST . . .

Lesson 7-1: Geometric Mean

re label the sides as lengths
Re-label the Sides (as lengths)

Lesson 7-1: Geometric Mean

geometric mean 1
Geometric Mean #1

What is the proportion that uses f?

f is the geometric mean of d and e.

Lesson 7-1: Geometric Mean

geometric mean 2
Geometric Mean #2

What is the proportion that uses b?

b is the geometric mean of e and c.

Lesson 7-1: Geometric Mean

geometric mean 3
Geometric Mean #3

What is the proportion that uses a?

a is the geometric mean of d and c.

Lesson 7-1: Geometric Mean

put them all together
Put them all together

Lesson 7-1: Geometric Mean

the w pattern
The “W” Pattern

W

Lesson 7-1: Geometric Mean

try it
Try it !

Given: d = 4 and e = 10

Find:

a = ___

b = ___

c = ___

f = ___

Lesson 7-1: Geometric Mean

solution
Solution:

Proportions

Answers

Lesson 7-1: Geometric Mean