1. Course 3 13-2 Terms of Geometric Sequences 13-2 Terms of Geometric Sequences Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Course 3 13-2 Terms of 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 Course 3 13-2 Terms of Geometric Sequences

3. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of Geometric Sequences Learn to find terms in a geometric sequence.

5. Course 3 13-2 Terms of Geometric Sequences Vocabulary geometric sequence common ratio

6. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 1, 5, 25, 125, 625, … Additional Example 1A: Identifying Geometric Sequences The sequence could be geometric with a common ratio of 5. Divide each term by the term before it. 1 5 25 125 625,... 5 5 55

8. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 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. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 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. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. A. 2, 10, 50, 250, 1250,... Check It Out: Example 1A The sequence could be geometric with a common ratio of 5. Divide each term by the term before it. 2 10 50 250 1250,... 5 5 55

12. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 1, 1, 1, 1, 1,... Check it Out: Example 1B The sequence could be geometric with a common ratio of 1. Divide each term by the term before it. 1 1 1 1 1,... 1 1 11

13. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 2, 4, 12, 24, 96,... Check It Out: Example 1C The sequence is not geometric. Divide each term by the term before it. 2 4 12 24 96,... 4 2 32

14. Course 3 13-2 Terms of Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. 1, 2, 4, 8, 16,... Check it Out: 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. Course 3 13-2 Terms of Geometric Sequences

16. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 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. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 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. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 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. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 12 th term: -2, 4, -8, 16,... Check It Out: Example 2A a n = a 1 r n–1 a 12 = –2(–2) 11 = –2(–2048) = 4096 r = = –2 4 –2

21. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 11 th term: 100, 70, 49, 34.3,... Check It Out: 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. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 5 th term: 0.01, 0.1, 1, 10,... Check It Out: 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. Course 3 13-2 Terms of Geometric Sequences Find the given term in the geometric sequence. 12 th term: 1000, 200, 40, 8, … Check It Out: 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. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of 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. Course 3 13-2 Terms of Geometric Sequences In a tennis tournament, there are 128 players competing in the first round. There are 64 players remaining in the second round, 32 players remaining in the third round, and so on. How many teams are remaining in the 5th round? Check It Out: Example 3 a 5 = (128)(0.5) 4 = (128)(0.0625) = 48 a n = a 1 r n–1 r = = 0.5 64 128 n = 5a 1 = 128 There will be 8 players remaining in the 5th round.

27. Course 3 13-2 Terms of 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