1. 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 Warm Up

2. Pre-Algebra 12-2 Geometric Sequences

3. Learn to find terms in a geometric sequence.

4. geometric sequence common ratio Vocabulary

5. 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.

6. Determine if the sequence could be geometric. If so, give the common ratio. A. 1, 5, 25, 125, 625, … 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 Example: Identifying Geometric Sequences

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

8. Determine if the sequence could be geometric. If so, give the common ratio. C. 81, 27, 9, 3, 1,... 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 Example: Identifying Geometric Sequences

9. Determine if the sequence could be geometric. If so, give the common ratio. D. –3, 6, –12, 24, –48 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 Example: Identifying Geometric Sequences

10. Determine if the sequence could be geometric. If so, give the common ratio. A. 2, 10, 50, 250, 1250,... 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 Try This

11. Determine if the sequence could be geometric. If so, give the common ratio. B. 1, 1, 1, 1, 1,... 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 Try This

12. Determine if the sequence could be geometric. If so, give the common ratio. C. 2, 4, 12, 24, 96,... The sequence is not geometric. Divide each term by the term before it. 2 4 12 24 96,... 4 2 32 Try This

13. Determine if the sequence could be geometric. If so, give the common ratio. D. 1, 2, 4, 8, 16,... 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. Try This

14. 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.

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

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

17. Find the given term in the geometric sequence. C. 10 th term: 0.01, 0.1, 1, 10,... 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 Example: Finding a Given Term of a Geometric Sequence

18. Find the given term in the geometric sequence. D. 7 th term: 1000, 200, 40, 8,... 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 Example: Finding a Given Term of a Geometric Sequence

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

20. Find the given term in the geometric sequence. B. 11 th term: 100, 70, 49, 34.3,... a 11 = 100(0.7) 10 = 100(0.0282475249)  2.825 a n = a 1 r n–1 r = = 0.7 70 100 Try This

21. Find the given term in the geometric sequence. C. 5 th term: 0.01, 0.1, 1, 10,... a 5 = 0.01(10) 4 = 0.01(10,000) = 100 a n = a 1 r n–1 r = = 10 0.1 0.01 Try This

22. Find the given term in the geometric sequence. D. 12 th term: 1000, 200, 40, 8, … 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 Try This

23. 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. Example: Money Application

24. 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. Example Continued

25. 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? 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. Try This

26. 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 Lesson Quiz