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

2. Pre-Algebra 12-2 Geometric Sequences 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. 12 th term: a 1 = 30, d = 0.5 3. 55 th term: 4, 28, 52, 76 no 35.5 1300 Pre-Algebra 12-2 Geometric Sequences

3. Pre-Algebra 12-2 Geometric Sequences 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. Pre-Algebra 12-2 Geometric Sequences Learn to find terms in a geometric sequence.

5. Pre-Algebra 12-2 Geometric Sequences Vocabulary geometric sequence common ratio

6. Pre-Algebra 12-2 Geometric Sequences 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. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. A. 1, 5, 25, 125, 625, … Additional Example 1A: Identifying Geometric Sequences The sequence could be a geometric with a common ratio of 5. Divide each term by the term before it. 1 5 25 125 625,... 5 5 55

8. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. B. 1, 3, 9, 12, 15, … Additional Example 1B: Identifying Geometric Sequences The sequence is not geometric. Divide each term by the term before it. 1 3 9 12 15,... 5454 4343 33

9. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. C. 81, 27, 9, 3, 1,... Additional Example 1C: Identifying Geometric Sequences The sequence could be geometric with a common ratio of. 1 3 Divide each term by the term before it. 81 27 9 3 1,... 1313 1313 1313 1313

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

11. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. A. 2, 10, 50, 250, 1250,... Try This: Example 1A The sequence could be a geometric with a common ratio of 5. Divide each term by the term before it. 2 10 50 250 1250,... 5 5 55

12. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. B. 1, 1, 1, 1, 1,... Try This: Example 1B The sequence could be a geometric with a common ratio of 1. Divide each term by the term before it. 1 1 1 1 1,... 1 1 11

13. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. C. 2, 4, 12, 24, 96,... Try This: Example 1C The sequence is not geometric. Divide each term by the term before it. 2 4 12 24 96,... 4 2 32

14. Pre-Algebra 12-2 Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. D. 1, 2, 4, 8, 16,... Try This: Example 1D The sequence could be geometric with a common ratio of 2. 1 2 4 8 16,... 2 2 22 Divide each term by the term before it.

15. Pre-Algebra 12-2 Geometric Sequences FINDING THE n th TERM OF A GEOMETRIC SEQUENCE The n th term an of a geometric sequence with common ratio r is a n = a 1 r n–1.

16. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. A. 11 th term: –2, 4, –8, 16,... Additional Example 2A: Finding a Given Term of a Geometric Sequence a n = a 1 r n–1 a 11 = –2(–2) 10 = –2(1024) = –2048 r = = –2 4 –2

17. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. B. 9 th term: 100, 70, 49, 34.3,... Additional Example 2B: Finding a Given Term of a Geometric Sequence a n = a 1 r n–1 a 9 = 100(0.7) 8 = 100(0.05764801) = 5.764801 r = = 0.7 70 100

18. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. C. 10 th term: 0.01, 0.1, 1, 10,... Additional Example 2C: Finding a Given Term of a Geometric Sequence a n = a 1 r n–1 a 10 = 0.01(10) 9 = 0.01(1,000,000,000) = 10,000,000 r = = 10 0.1 0.01

19. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. D. 7 th term: 1000, 200, 40, 8,... Additional Example 2D: Finding a Given Term of a Geometric Sequence a n = a 1 r n–1 r = = 200 1000 1 5 a 7 = 1000 ( ) 6 = 1000 ( ) =, or 0.064 1 5 8 125 1 15,625

20. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. A. 12 th term: -2, 4, -8, 16,... Try This: Example 2A a n = a 1 r n–1 a 12 = –2(–2) 11 = –2(–2048) = 4096 r = = –2 4 –2

21. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. B. 11 th term: 100, 70, 49, 34.3,... Try This: Example 2B a 11 = 100(0.7) 10 = 100(0.0282475249)  2.825 a n = a 1 r n–1 r = = 0.7 70 100

22. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. C. 5 th term: 0.01, 0.1, 1, 10,... Try This: Example 2C a 5 = 0.01(10) 4 = 0.01(10,000) = 100 a n = a 1 r n–1 r = = 10 0.1 0.01

23. Pre-Algebra 12-2 Geometric Sequences Find the given term in the geometric sequence. D. 12 th term: 1000, 200, 40, 8, … Try This: Example 2D a n = a 1 r n–1 r = = 200 1000 1 5 a 5 = 1000 ( ) 4 = 1000 ( ) =, or 1.6 1 5 8 5 1 625

24. Pre-Algebra 12-2 Geometric Sequences 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? Additional Example 3: Money Application If Tara chooses $50 per sale, she will get a total of 10($50) = $500.

25. Pre-Algebra 12-2 Geometric Sequences Additional Example 3 Continued a 10 = ($1)(2) 9 = ($1)(512) = $512 Option 1 gives Tara more money in the beginning, but option 2 gives her a larger total amount. If Tara chooses the second option, her earnings for just the 10 th sale will be more that the total of all the earnings in option 1.

26. Pre-Algebra 12-2 Geometric Sequences 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? Try This: Example 3 a 7 = (932)(0.98) 6  (932)(0.89)  829 a n = a 1 r n–1 r =  0.98 913 932 n = 7 a 1 = 932 There will be about 829 gumballs in the machine after 7 days.

27. Pre-Algebra 12-2 Geometric Sequences 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. 7 th term:, 1, 3, 9,... 4. 20 th term: a 1 = 800, r = 0.8 no 243 ≈ 11.53 1 3 yes; 1 2