1. Check 12-1 HOMEWORK

2. Pre-Algebra HOMEWORK
Page 606 #10-18

3. Students will be able to solve sequences and represent functions by completing the following assignments.
Learn to find terms in an arithmetic sequence.
Learn to find terms in a geometric sequence.
Learn to find patterns in sequences.
Learn to represent functions with tables, graphs, or equations.

4. Today’s Learning Goal Assignment
Learn to find terms in a geometric sequence.

5. Geometric Sequences 12-2 Warm Up Problem of the Day
Lesson Presentation
Pre-Algebra

6. Geometric Sequences 12-2 Warm Up
Pre-Algebra
12-2
Geometric Sequences
Warm Up
1. Determine if the sequence could be arithmetic. If so, give the common difference , 50, 25, 12.5, . . .
Find the given term in each arithmetic sequence.
2. 12th term: a1 = 30, d = 0.5
3. 55th term: 4, 28, 52, 76 no
35.5
1300

7. Problem of the Day
Two students begin counting by 3’s at the same time. One counts up from 0, and the other counts down from 120. If each says one number every second, will both students ever say the same number at the same time?
yes (60)

8. Vocabulary
geometric sequence
common ratio

9. In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio. The common ratio is multiplied by each term to get the next term.

10. Additional Example 1A: Identifying Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
A. 1, 5, 25, 125, 625, …
Divide each term by the term before it.
, . . . 5 5 5 5
The sequence could be a geometric with a common ratio of 5.

11. Try This: Example 1A
Determine if the sequence could be geometric. If so, give the common ratio.
A. 2, 10, 50, 250, 1250, . . .
Divide each term by the term before it.
, . . . 5 5 5 5
The sequence could be a geometric with a common ratio of 5.

12. Additional Example 1B: Identifying Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 3, 9, 12, 15, …
Divide each term by the term before it.
, . . . 3 3 43 54
The sequence is not geometric.

13. Try This: Example 1B
Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 1, 1, 1, 1, . . .
Divide each term by the term before it.
, . . . 1 1 1 1
The sequence could be a geometric with a common ratio of 1.

14. Additional Example 1C: Identifying Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
C. 81, 27, 9, 3, 1, . . .
Divide each term by the term before it.
, . . . 13 13 13 13
The sequence could be geometric with a common ratio of . 1 3

15. Try This: Example 1C
Determine if the sequence could be geometric. If so, give the common ratio.
C. 2, 4, 12, 24, 96, . . .
Divide each term by the term before it.
, . . . 2 3 2 4
The sequence is not geometric.

16. Additional Example 1D: Identifying Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
D. –3, 6, –12, 24, –48
Divide each term by the term before it.
– – –48, . . . –2 –2 –2 –2
The sequence could be geometric with a common ratio of –2.

17. Try This: Example 1D
Determine if the sequence could be geometric. If so, give the common ratio.
D. 1, 2, 4, 8, 16, . . .
Divide each term by the term before it.
, . . . 2 2 2 2
The sequence could be geometric with a common ratio of 2.

18. FINDING THE nth TERM OF A GEOMETRIC SEQUENCE
The nth term an of a geometric sequence with common ratio r is
an = a1rn–1.

19. Additional Example 2A: Finding a Given Term of a Geometric Sequence
Find the given term in the geometric sequence.
A. 11th term: –2, 4, –8, 16, . . .
r = = –2 4 –2
an = a1rn–1
a11 = –2(–2)10 = –2(1024) = –2048

20. Try This: Example 2A
Find the given term in the geometric sequence.
A. 12th term: -2, 4, -8, 16, . . .
r = = –2 4 –2
an = a1rn–1
a12 = –2(–2)11 = –2(–2048) = 4096

21. Additional Example 2B: Finding a Given Term of a Geometric Sequence
Find the given term in the geometric sequence.
B. 9th term: 100, 70, 49, 34.3, . . .
r = = 0.7 70
100
an = a1rn–1
a9 = 100(0.7)8 = 100( ) =

22. Try This: Example 2B
Find the given term in the geometric sequence.
B. 11th term: 100, 70, 49, 34.3, . . .
r = = 0.7 70
100
an = a1rn–1
a11 = 100(0.7)10 = 100( )  2.825

23. Additional Example 2C: Finding a Given Term of a Geometric Sequence
Find the given term in the geometric sequence.
C. 10th term: 0.01, 0.1, 1, 10, . . .
r = = 10
0.1
0.01
an = a1rn–1
a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000

24. Try This: Example 2C
Find the given term in the geometric sequence.
C. 5th term: 0.01, 0.1, 1, 10, . . .
r = = 10
0.1
0.01
an = a1rn–1
a5 = 0.01(10)4 = 0.01(10,000) = 100

25. Additional Example 2D: Finding a Given Term of a Geometric Sequence
Find the given term in the geometric sequence.
D. 7th term: 1000, 200, 40, 8, . . .
r = =
200
1000 1 5
an = a1rn–1
a7 = 1000( )6 = 1000( )= , or 0.064 1 5 8
125
15,625

26. Try This: Example 2D
Find the given term in the geometric sequence.
D. 12th term: 1000, 200, 40, 8, …
r = =
200
1000 1 5
an = a1rn–1
a5 = 1000 ( )4 = 1000( )= , or 1.6 1 5 8
625

27. Additional Example 3: Money Application
Tara sells computers. She has the option of earning (1) $50 per sale or (2) $1 for the first sale, $2 for the second sale, $4 for the third sale and so on, where each sale is worth twice as much as the previous sale. If Tara estimates that she can sell 10 computers a week, which option should she choose?
If Tara chooses $50 per sale, she will get a total of 10($50) = $500.

28. Additional Example 3 Continued
If Tara chooses the second option, her earnings for just the 10th sale will be more that the total of all the earnings in option 1.
a10 = ($1)(2)9 = ($1)(512) = $512
Option 1 gives Tara more money in the beginning, but option 2 gives her a larger total amount.

29. Try This: Example 3
A gumball machine at the mall has 932 gumballs. If 19 gumballs are bought each day, how many gumballs will be left in the machine on the 7th day?
n = 7
a1 = 932
r =  0.98
913
932
an = a1rn–1
a7 = (932)(0.98)6  (932)(0.89)  829
There will be about 829 gumballs in the machine after 7 days.

30. Lesson Quiz
Determine if each sequence could be geometric. If so, give the common ratio.
1. 200, 100, 50, 25, 12.5, . . .
2. 4, 8, 12, 16, . . .
Find the given term in each geometric sequence.
3. 7th term: , 1, 3, 9, . . .
4. 20th term: a1 = 800, r = 0.8
yes; 1 2 no 1 3
243
≈ 11.53