2, 4, 8, 16, …

1. Exercise 32 2, 4, 8, 16, …

2. Exercise 10 2, 4, 6, 8, …

3. Exercise 81 1, 3, 9, 27, …

4. 116 Exercise 18 12 14 1, , , , …

5. Exercise –32 1, –2, 4, –8, 16, …

6. 6 12 24 48 96 x 2 x 2 x 2 x 2 x 2 3

7. Geometric Sequence A geometric sequence is a sequence of numbers whose successive terms differ by a constant multiplier.

8. Common Ratio The constant multiplier for a geometric sequence is called the common ratio, r.

9. Example State whether the sequence 8, 4, 2, 1, … is arithmetic or geometric. geometric

10. Example State whether the sequence –6, –18, –54, –162, … is arithmetic or geometric. geometric

11. Example State whether the sequence 5, 7, 9, 11, … is arithmetic or geometric. arithmetic

12. Example State whether the sequence 5, 10, 20, 40, … is arithmetic or geometric. geometric

13. Geometric Sequence Terms differ by a constant factor r. an = an – 1r

14. Example 1 Write the first six terms of the geometric sequence in which a1 = 1 and r = 3. a1 = 1 a2 = 1 • 3 = 3 a3 = 3 • 3 = 9 a4 = 9 • 3 = 27

15. Example 1 Write the first six terms of the geometric sequence in which a1 = 1 and r = 3. a5 = 27 • 3 = 81 a6 = 81 • 3 = 243 The first six terms of the sequence are 1, 3, 9, 27, 81, and 243.

16. Example 2 Find the value of a1 for the sequence 2, 6, 18, 54, 162, 486, … a1 = 2

17. Example 2 Find the value of r for the sequence 2, 6, 18, 54, 162, 486, … r = 3

18. Example 2 Find the value of a3 for the sequence 2, 6, 18, 54, 162, 486, … a3 = 18

19. Example 2 Find the value of a8 for the sequence 2, 6, 18, 54, 162, 486, … a7 = 486 • 3 = 1,458 a8 = 1,458 • 3 = 4,374

20. Geometric Sequence Terms differ by a constant factor r. an = an – 1r

21. Example 3 Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1. a1 = –4 a2 = 3(–4) = –12 a3 = 3(–12) = –36 a4 = 3(–36) = –108 a5 = 3(–108) = –324

22. Example 3 Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1. The first five terms of the sequence are –4, –12, –36, –108, and –324.

23. Example Write the first four terms of the sequence defined bya1 = 2 and an = 4an – 1. 2, 8, 32, 128

24. Example Write the first four terms of the sequence defined bya1 = –3 and an = 2an – 1. –3, –6, –12, –24

25. Example Find the common ratio, r, of the sequence 4, –12, 36, –108. r = –3

26. 12 r = Example Find the common ratio, r, of the sequence 24, 12, 6, 3.

27. 13 r = 13 an = an – 1 Example 4 Write the recursive formula for the sequence 729, 243, 81, 27, ... a1 = 729

28. 3, 6 ,12, 24, 48, 96 × 2 to get next term

29. Position Term 1 3 3 • 21 = 6 2 3 3 • 22 = 12 3 • 23 = 24 4 3 • 24 = 48 5 3 • 25 = 96 6 3 • 2n – 1 n

30. Explicit Formula The explicit formula for a geometric sequence is an = a1rn–1, in which a1 is the first term and r is the common ratio.

31. Example 5 Write the explicit formula for the sequence –5, –15, –45, –135, –405, ... a1 = –5 r =3 an = –5(3)n– 1

32. Example Write the explicit formula for the sequence –3, –6, –12, –24, ... an = –3(2)n– 1

33. 12 an = 12( )n – 1 Example Write the explicit formula for the sequence 12, 6, 3, 1.5, ...

34. Exercise A ball bounces three-fourths the height of its fall. If the ball falls 12 ft., how high does it bounce on the first bounce? on the second bounce? on the third bounce? 9 ft.; 6.75 ft.; 5.0625 ft.

35. Exercise In the last problem, the height of the bounces forms a geometric sequence. Find the common ratio of this geometric sequence. r = 0.75

36. Exercise If the ball falls 12 ft. and begins bouncing, what is the total distance it has traveled when it hits the ground the third time? 43.5 ft.

37. Exercise When will the ball stop bouncing?