Lesson 7-1: Geometric Mean 1 Geometric Mean Lesson 7-1.
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Transcript of Lesson 7-1: Geometric Mean 1 Geometric Mean Lesson 7-1.
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