Geometric Mean

Guess The Next Term

Guess The Next Term 2, 6, 18, 54,____ 162 4, 4, 8, 12, 20,____ 32 3, 12, 36, 45, 135, ____ 144

Arithmetic Sequence , __ , __ , __ 2, 5, 8, 11, 14 17 20 23 The following pattern is called an arithmetic sequence: What are the next three terms? , __ , __ , __ 2, 5, 8, 11, 14 17 20 23 Describe this pattern?

Geometric Sequence 2, 6, 18, 54, 162 , ___, ___, ___ 486 1458 4374 The following pattern is called an geometric sequence: 2, 6, 18, 54, 162 What are the three missing terms? , ___, ___, ___ 486 1458 4374 Describe this pattern?

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

Example Starting with the number 1 and using a factor of 4, create 5 terms of a geometric sequence. 1 , 4 , 16 , 64 , 256

Try it #1 2 , 10 , 50 , 250 , 1250 Starting with the number 2 and using a factor of 5, create 5 terms of a geometric sequence. 2 , 10 , 50 , 250 , 1250

Try it #2 5 , 15 , 45 , 135 , 405 Starting with the number 5 and using a factor of 3, create 5 terms of a geometric sequence. 5 , 15 , 45 , 135 , 405

Try it #3 2 , ___ , 72 , 432 12 The following is a geometric sequence. Find the missing term. 2 , ___ , 72 , 432 12

Try it #4 6 , ___ , 24 12 The following is a geometric sequence. Find the missing term. 6 , ___ , 24 12

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

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

Try it #5 Find the geometric mean of 3 and 300. 3 , ___ , 300 30

Fact Consecutive terms of a geometric sequence are proportional.

Example Geometric Sequence (with a factor of 10): 4 , 40 , 400 cross-products are equal (4)(400) = (40)(40) 1600 = 1600

Therefore 7 , ___ , 28 To find the geometric mean between 7 and 28 ... label the missing term x write a proportion cross multiply solve

Try it ! Find the geometric mean of . . . Answer = 20 1. 10 and 40

How does this relate to geometry?

The " W " Pattern

The Geometric Means Recall the three geometric means that you discovered from your Sketchpad activity. BUT FIRST . . .

Re-label the Sides (as lengths)

Geometric Mean #1 f is the geometric mean of d and e. What is the proportion that uses f? f is the geometric mean of d and e.

Geometric Mean #2 b is the geometric mean of e and c. What is the proportion that uses b? b is the geometric mean of e and c.

Geometric Mean #3 a is the geometric mean of d and c. What is the proportion that uses a? a is the geometric mean of d and c.

Put them all together

The “W” Pattern W

Try it ! d = 4 e = 10 Find: a = ___ b = ___ c = ___ f = ___

Solution: proportions answers