1. 2, 4, 8, 16, … 32 Exercise

2. 2, 4, 6, 8, … Exercise 10

3. 1, 3, 9, 27, … 81 Exercise

4. 1,,,, … 1212 1212 1414 1414 1818 1818 1 16 Exercise

5. 1, –2, 4, –8, 16, … –32 Exercise

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

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

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

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

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

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

14. Write the first six terms of the geometric sequence in which a 1 = 1 and r = 3. a 1 = 1 a 2 = 1 3 = 3 a 3 = 3 3 = 9 a 4 = 9 3 = 27 Example 1

15. The first six terms of the sequence are 1, 3, 9, 27, 81, and 243. Write the first six terms of the geometric sequence in which a 1 = 1 and r = 3. Example 1 a 5 = 27 3 = 81 a 6 = 81 3 = 243

16. Find the value of a 1 for the sequence 2, 6, 18, 54, 162, 486, … a 1 = 2 Example 2

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

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

19. Find the value of a 8 for the sequence 2, 6, 18, 54, 162, 486, … = 4,374 a 7 = 486 3 = 1,458 a 8 = 1,458 3 Example 2

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

21. Write the first five terms of the sequence defined by a 1 = –4 and a n = 3a n – 1. a 1 = –4 a 2 = 3(–4) = –12 a 3 = 3(–12) = –36 a 4 = 3(–36) = –108 a 5 = 3(–108) = –324 Example 3

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

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

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

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

26. Find the common ratio, r, of the sequence 24, 12, 6, 3. r = 1212 1212 Example

27. Write the recursive formula for the sequence 729, 243, 81, 27,... a 1 = 729 r = 1313 1313 a n = a n – 1 1313 1313 Example 4

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

29. 1 Position Term 2 3 4 5 6 n 3 3 2 1 = 6 3 2 2 = 12 3 2 3 = 24 3 2 4 = 48 3 2 5 = 96 3 2 n – 1

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

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

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

33. Write the explicit formula for the sequence 12, 6, 3, 1.5,... a n = 12( ) n – 1 1212 1212 Example

34. 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. Exercise

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

36. 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. Exercise

37. When will the ball stop bouncing? Exercise