Lesson 7-1: Geometric Mean 1

Geometric Mean

Lesson 7-1

Lesson 7-1: Geometric Mean 2

Sequences

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.

Arithmetic Sequence:

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, 5, 8, 11, 14, ____, ____, ____17 20 23 Common difference = 3

Example: 2, 6, 18, 54, 162, _____, _____, ____486 1458 4374 Common Factor = 3

Lesson 7-1: Geometric Mean 3

Examples1. 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

5. In the geometric sequence 6, ____, 24,... Find the missing term.12

Lesson 7-1: Geometric Mean 4

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

Example:

Find the geometric mean of 3 and 300.

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

Try It:

3 , ___ , 30030

Lesson 7-1: Geometric Mean 5

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

40 =

40

400

cross-products are equal (4)(400) = (40)(40)

1600 = 1600

Lesson 7-1: Geometric Mean 6

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

7 , ___ , 28label the missing term x

write a proportion

cross multiply

solve

7X

= X28

X

X2 = (7)(28) X2 = 196

X2 = 196 X = 14

Lesson 7-1: Geometric Mean 7

a x

x b ab

The geometric mean between two numbers a and b is the

positive number x where . Therefore x = .

Try It: Find the geometric mean of . . .

1. 10 and 40 Answer = 20

2. 1 and 36 Answer = 6

3. 10 and 20 Answer = 14.14

4. 5 and 6 Answer = 5.48

5. 8.1 and 12.2 Answer = 9.94

Lesson 7-1: Geometric Mean 8

How does this relate to geometry?

D CB

A

Lesson 7-1: Geometric Mean 9

D CB

A

Lesson 7-1: Geometric Mean 10

The Geometric Means

D CB

A

Recall the three geometric means that you discovered from your Sketchpad activity.

BUT FIRST . . .

Lesson 7-1: Geometric Mean 11

Re-label the Sides (as lengths)

a f

d

ba

c e

fb

a bf

d ec

Lesson 7-1: Geometric Mean 12

a bf

d e

Geometric Mean #1

af

d e

fb

d

f =

f

e

f is the geometric mean of d and e.

What is the proportion that uses f?

Lesson 7-1: Geometric Mean 13

Geometric Mean #2

ba

c e

fb

a bf

d ec

e

b =

b

c

b is the geometric mean of e and c.

What is the proportion that uses b?

Lesson 7-1: Geometric Mean 14

Geometric Mean #3

d

a =

a

c

ba

c

af

d

a bf

d ec

a is the geometric mean of d and c.

What is the proportion that uses a?

Lesson 7-1: Geometric Mean 15

Put them all together

a bf

d ec

d

a =

a

c

e

b =

b

cd

f =

f

e

Lesson 7-1: Geometric Mean 16

The “W” Pattern

a bf

d ecW

Lesson 7-1: Geometric Mean 17

Try it !

a bf

d ec

Given: d = 4 and e = 10

Find:a = ___b = ___c = ___f = ___

Lesson 7-1: Geometric Mean 18

Solution:

4

f =

f

10

4

a =

a

1410

b =

b

14

a bf

4 1014

a = 7.48 f = 6.32 b = 11.83

Proportions

Answers