12-2

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

2. Geometric Sequences 12-2 Pre-Algebra Warm Up 1.Determine if the sequence could be arithmetic. If so, give the common difference. 100, 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

3. 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)

4. Learn to find terms in a geometric sequence.

5. Vocabulary geometric sequence common ratio

6. 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.

7. 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. 1 5 25 125 625, . . . 5 5 5 5 The sequence could be a geometric with a common ratio of 5.

8. 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. 1 3 9 12 15, . . . 54 43 3 3 The sequence is not geometric.

9. The sequence could be geometric with a common ratio of . 1 3 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. 81 27 9 3 1, . . . 13 13 13 13

10. 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. –3 6 –12 24 –48, . . . –2 –2 –2 –2 The sequence could be geometric with a common ratio of –2.

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. 2 10 50 250 1250, . . . 5 5 5 5 The sequence could be a geometric with a common ratio of 5.

12. 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 1, . . . 1 1 1 1 The sequence could be a geometric with a common ratio of 1.

13. 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 4 12 24 96, . . . 4 2 2 3 The sequence is not geometric.

14. 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. 1 2 4 8 16, . . . 2 2 2 2 The sequence could be geometric with a common ratio of 2.

16. r = = –2 4 –2 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, . . . an = a1rn–1 a11 = –2(–2)10 = –2(1024) = –2048

17. r = = 0.7 70 100 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, . . . an = a1rn–1 a9 = 100(0.7)8 = 100(0.05764801) = 5.764801

18. 0.1 r = = 10 0.01 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, . . . an = a1rn–1 a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000

19. 200 r = = 1000 1 1 5 5 a7 = 1000( )6 = 1000( )= , or 0.064 8 1 125 15,625 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, . . . an = a1rn–1

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

21. r = = 0.7 70 100 Try This: Example 2B Find the given term in the geometric sequence. B. 11th term: 100, 70, 49, 34.3, . . . an = a1rn–1 a11 = 100(0.7)10 = 100(0.0282475249)  2.825

22. 0.1 r = = 10 0.01 Try This: Example 2C Find the given term in the geometric sequence. C. 5th term: 0.01, 0.1, 1, 10, . . . an = a1rn–1 a5 = 0.01(10)4 = 0.01(10,000) = 100

23. 200 r = = 1000 1 1 5 5 a5 = 1000 ( )4= 1000( )= , or 1.6 1 8 625 5 Try This: Example 2D Find the given term in the geometric sequence. D. 12th term: 1000, 200, 40, 8, … an = a1rn–1

24. 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.

25. 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.

26. 913 r =  0.98 932 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? a1 = 932 n = 7 an = a1rn–1 a7 = (932)(0.98)6 (932)(0.89)  829 There will be about 829 gumballs in the machine after 7 days.

27. 1 3 yes; 1 2 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 no 243 ≈ 11.53