Geometric Mean and Radicals
 Keystone Geometry
2
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.
Common Factor = 3
Example: 2, 6, 18, 54, 162, _____,
486 1458
_____, 4374
____
3
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
12 72, 432…
4. Find the missing term in the geometric sequence 2, ____,
12 24,...
5. Find the missing term in the geometric sequence 6, ____,
4
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
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
cross-products are equal
=
40
400
(4)(400) = (40)(40)
1600 = 1600
6
Therefore ………..
To find the geometric mean between 7 and 28 ...
7 , ___
X , 28
label the missing term x
7
X
=
X 28
write a proportion
cross multiply
solve
X2 = (7)(28)
X2 = 196
X2 = 196
X = 14
7
The geometric mean between two numbers a and b is the
positive number x where a = x . Therefore x =
x
b
ab
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