Transcript Lesson 7-1 Geometric Mean Lesson 7-1: Geometric Mean Sequences Arithmetic Sequence: Is a pattern of numbers where any term (number in the sequence) is determined.

Lesson 7-1
Geometric Mean
Lesson 7-1: Geometric Mean
1
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 Common difference = 3
Example: 2, 5, 8, 11, 14, ____,
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
2
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, ____,
12 72, 432, .Find the missing term.
12 24,... Find the missing term.
5. In the geometric sequence 6, ____,
Lesson 7-1: Geometric Mean
3
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 , ___
30 , 300
Lesson 7-1: Geometric Mean
4
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
cross-products are equal
=
40
400
(4)(400) = (40)(40)
1600 = 1600
Lesson 7-1: Geometric Mean
5
Therefore ………..
To find the geometric mean between 7 and 28 ...
X , 28
7 , ___
label the missing term x
7
X
=
X 28
write a proportion
cross multiply
solve
X2 = (7)(28)
X2 = 196
X2 = 196
X = 14
Lesson 7-1: Geometric Mean
6
The geometric mean between two numbers a and b is the
a x

ab .
positive number x where x b . 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
7
How does this relate to geometry?
A
B
C
D
Lesson 7-1: Geometric Mean
8
A
B
D
Lesson 7-1: Geometric Mean
C
9
The Geometric Means
A
B
D
C
Recall the three geometric means that you
discovered from your Sketchpad activity.
BUT FIRST . . .
Lesson 7-1: Geometric Mean
10
Re-label the Sides (as lengths)
a
f
d
c
a
f
d
b
e
b
b
a
f
c
Lesson 7-1: Geometric Mean
e
11
Geometric Mean #1
a
b
f
f
d
e
What is the proportion that uses f?
a
d
d
f
=
f
e
b
f
f is the geometric mean of d and e.
e
Lesson 7-1: Geometric Mean
12
Geometric Mean #2
a
b
c
b
f
e
What is the proportion that uses b?
a
d
c
b
b
f
e
=
b
c
b is the geometric mean of e and c.
e
Lesson 7-1: Geometric Mean
13
Geometric Mean #3
a
f
a
b
c
d
What is the proportion that uses a?
a
c
a
=
a
c
b
f
d
d
a is the geometric mean of d and c.
e
Lesson 7-1: Geometric Mean
14
Put them all together
a
f
d
c
d
a
=
a
c
b
d
f
=
e
f
e
e
b
Lesson 7-1: Geometric Mean
=
15
b
c
The “W” Pattern
W
a
f
d
c
b
e
Lesson 7-1: Geometric Mean
16
Try it !
Given: d = 4 and e = 10
Find:
a = ___
b = ___
c = ___
f = ___
a
b
f
d
c
Lesson 7-1: Geometric Mean
e
17
Solution:
a
a
=
f
4
Proportions
4
b
a
14
Answers
a = 7.48
10
14
4
f
=
f
10
10
b
f = 6.32
Lesson 7-1: Geometric Mean
=
b
14
b = 11.83
18