Pre-Algebra

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

2. Geometric Sequences 12-2 Pre-Algebra 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.

3. Geometric Sequences 12-2 Pre-Algebra 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.

4. Geometric Sequences 12-2 Pre-Algebra 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.

5. The sequence could be geometric with a common ratio of . Geometric Sequences 12-2 1 3 Pre-Algebra 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

6. Geometric Sequences 12-2 Pre-Algebra 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.

7. Geometric Sequences 12-2 Pre-Algebra

8. 4 Geometric Sequences 12-2 r = = –2 –2 Pre-Algebra 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

9. 70 Geometric Sequences 12-2 r = = 0.7 100 Pre-Algebra 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

10. 0.1 Geometric Sequences 12-2 r = = 10 0.01 Pre-Algebra 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

11. 200 1 1 1 a7 = 1000( )6 = 1000( )= , or 0.064 8 Geometric Sequences 12-2 r = = 1000 125 5 15,625 5 Pre-Algebra 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

12. Geometric Sequences 12-2 Pre-Algebra 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.

13. Geometric Sequences 12-2 Pre-Algebra 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.

14. Geometric Sequences 12-2 Pre-Algebra 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.

15. Geometric Sequences 12-2 Pre-Algebra

16. 4 Geometric Sequences 12-2 r = = –2 –2 Pre-Algebra 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

17. 70 Geometric Sequences 12-2 r = = 0.7 100 Pre-Algebra 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

18. Geometric Sequences 12-2 Pre-Algebra Example 3: Money Application Siobhan 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 Siobhan estimates that she can sell 10 computers a week, which option should she choose? If Siobhan chooses $50 per sale, she will get a total of 10($50) = $500.

19. Geometric Sequences 12-2 Pre-Algebra Example 3 Continued If Siobhan 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 Siobhan more money in the beginning, but option 2 gives her a larger total amount.

20. Geometric Sequences 12-2 Pre-Algebra 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.

21. Geometric Sequences 12-2 Pre-Algebra

22. Geometric Sequences 12-2 Pre-Algebra 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.

23. Geometric Sequences 12-2 Pre-Algebra 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.

24. 0.1 Geometric Sequences 12-2 r = = 10 0.01 Pre-Algebra 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

25. 8 200 1 1 1 a5 = 1000 ( )4= 1000( )= , or 1.6 Geometric Sequences 12-2 r = = 1000 5 5 625 5 Pre-Algebra Try This: Example 2D Find the given term in the geometric sequence. D. 12th term: 1000, 200, 40, 8, … an = a1rn–1

26. 913 Geometric Sequences 12-2 r = ≈ 0.98 932 Pre-Algebra 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