Problem Set 39: Geometric Sequences

1. What is a geometric sequence?

2. How is the common ratio of a geometric sequence found?

3. What is the procedure for determining whether a sequence is geometric?

4. What is the difference between an arithmetic sequence and a geometric sequence?

5. Describe how exponential functions and geometric sequences are similar. How are they different?

For the following exercises, find the common ratio for the geometric sequence.

6. 1,3,9,27,81,..1,3,9,27,81,...

7. −0.125,0.25,−0.5,1,−2,..−0.125,0.25,−0.5,1,−2,...

8. −2,−12,−18,−132,−1128,..−2,−12,−18,−132,−1128,...

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

9. −6,−12,−24,−48,−96,..−6,−12,−24,−48,−96,...

10. 5,5.2,5.4,5.6,5.8,..5,5.2,5.4,5.6,5.8,...

11. −1,12,−14,18,−116,..−1,12,−14,18,−116,...

12. 6,8,11,15,20,..6,8,11,15,20,...

13. 0.8,4,20,100,500,..0.8,4,20,100,500,...

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

14. a1=8,r=0.3a1=8,r=0.3

15. a1=5,r=15a1=5,r=15

For the following exercises, write the first five terms of the geometric sequence, given any two terms.

16. a7=64,a10=512a7=64,a10=512

17. a6=25,a8=6.25a6=25,a8=6.25

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

18. The first term is 22, and the common ratio is 33. Find the 5th term.

19. The first term is 16 and the common ratio is −13−13. Find the 4th term.

For the following exercises, find the specified term for the geometric sequence, given the first four terms.

20. an={−1,2,−4,8,...}an={−1,2,−4,8,...}. Find a12a12.

21. an={−2,23,−29,227,...}an={−2,23,−29,227,...}. Find a7a7.

For the following exercises, write the first five terms of the geometric sequence.

22. a1=−486,an=−13an−1a1=−486,an=−13an−1

23. a1=7,an=0.2an−1a1=7,an=0.2an−1

For the following exercises, write a recursive formula for each geometric sequence.

24. an={−1,5,−25,125,...}an={−1,5,−25,125,...}

25. an={−32,−16,−8,−4,...}an={−32,−16,−8,−4,...}

26. an={14,56,224,896,...}an={14,56,224,896,...}

27. an={10,−3,0.9,−0.27,...}an={10,−3,0.9,−0.27,...}

28. an={0.61,1.83,5.49,16.47,...}an={0.61,1.83,5.49,16.47,...}

29. an={35,110,160,1360,...}an={35,110,160,1360,...}

30. an={−2,43,−89,1627,...}an={−2,43,−89,1627,...}

31. an={1512,−1128,132,−18,...}an={1512,−1128,132,−18,...}

For the following exercises, write the first five terms of the geometric sequence.

32. an=−4⋅5n−1an=−4⋅5n−1

33. an=12⋅(−12)n−1an=12⋅(−12)n−1

For the following exercises, write an explicit formula for each geometric sequence.

34. an={−2,−4,−8,−16,...}an={−2,−4,−8,−16,...}

35. an={1,3,9,27,...}an={1,3,9,27,...}

36. an={−4,−12,−36,−108,...}an={−4,−12,−36,−108,...}

37. an={0.8,−4,20,−100,...}

38. an={−1.25,−5,−20,−80,...}

39. an={−1,−45,−1625,−64125,...}

40. an={2,13,118,1108,...}

41. an={3,−1,13,−19,...}

For the following exercises, find the specified term for the geometric sequence given.

42. Let a1=4, an=−3an−1. Find a8.

43. Let an=−(−13)n−1. Find a12.

For the following exercises, find the number of terms in the given finite geometric sequence.

44. an={−1,3,−9,...,2187}

45. an={2,1,12,...,11024}

For the following exercises, determine whether the graph shown represents a geometric sequence.

46.
Graph of a scattered plot with labeled points: (1, -3), (2, -1), (3, 1), (4, 3), and (5, 5). The x-axis is labeled n and the y-axis is labeled a_n.

47.
Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

48. a1=1,r=12

49. a1=3,an=2an−1

50. an=27⋅0.3n−1

51. Use recursive formulas to give two examples of geometric sequences whose 3rd terms are 200.

52. Use explicit formulas to give two examples of geometric sequences whose 7th terms are 1024.

53. Find the 5th term of the geometric sequence {b,4b,16b,...}.

54. Find the 7th term of the geometric sequence {64a(−b),32a(−3b),16a(−9b),...}.

55. At which term does the sequence {10,12,14.4,17.28, ...} exceed 100?

56. At which term does the sequence {12187,1729,1243,181 ...} begin to have integer values?

57. For which term does the geometric sequence an=−36(23)n−1 first have a non-integer value?

58. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10th term.

59. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8th term.

60. Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.