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. [latex]1,3,9,27,81,..[/latex].

7. [latex]-0.125,0.25,-0.5,1,-2,..[/latex].

8. [latex]-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128},..[/latex].

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

9. [latex]-6,-12,-24,-48,-96,..[/latex].

10. [latex]5,5.2,5.4,5.6,5.8,..[/latex].

11. [latex]-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},..[/latex].

12. [latex]6,8,11,15,20,..[/latex].

13. [latex]0.8,4,20,100,500,..[/latex].

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

14. [latex]\begin{array}{cc}{a}_{1}=8,& r=0.3\end{array}[/latex]

15. [latex]\begin{array}{cc}{a}_{1}=5,& r=\frac{1}{5}\end{array}[/latex]

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

16. [latex]\begin{array}{cc}{a}_{7}=64,& {a}_{10}\end{array}=512[/latex]

17. [latex]\begin{array}{cc}{a}_{6}=25,& {a}_{8}\end{array}=6.25[/latex]

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

18. The first term is [latex]2[/latex], and the common ratio is [latex]3[/latex]. Find the 5^th term.

19. The first term is 16 and the common ratio is [latex]-\frac{1}{3}[/latex]. Find the 4^th term.

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

20. [latex]{a}_{n}=\left\{-1,2,-4,8,…\right\}[/latex]. Find [latex]{a}_{12}[/latex].

21. [latex]{a}_{n}=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},…\right\}[/latex]. Find [latex]{a}_{7}[/latex].

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

22. [latex]\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\frac{1}{3}\end{array}{a}_{n - 1}[/latex]

23. [latex]\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\end{array}[/latex]

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

24. [latex]{a}_{n}=\left\{-1,5,-25,125,…\right\}[/latex]

25. [latex]{a}_{n}=\left\{-32,-16,-8,-4,…\right\}[/latex]

26. [latex]{a}_{n}=\left\{14,56,224,896,…\right\}[/latex]

27. [latex]{a}_{n}=\left\{10,-3,0.9,-0.27,…\right\}[/latex]

28. [latex]{a}_{n}=\left\{0.61,1.83,5.49,16.47,…\right\}[/latex]

29. [latex]{a}_{n}=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},…\right\}[/latex]

30. [latex]{a}_{n}=\left\{-2,\frac{4}{3},-\frac{8}{9},\frac{16}{27},…\right\}[/latex]

31. [latex]{a}_{n}=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},…\right\}[/latex]

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

32. [latex]{a}_{n}=-4\cdot {5}^{n - 1}[/latex]

33. [latex]{a}_{n}=12\cdot {\left(-\frac{1}{2}\right)}^{n - 1}[/latex]

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

34. [latex]{a}_{n}=\left\{-2,-4,-8,-16,…\right\}[/latex]

35. [latex]{a}_{n}=\left\{1,3,9,27,…\right\}[/latex]

36. [latex]{a}_{n}=\left\{-4,-12,-36,-108,…\right\}[/latex]

37. [latex]{a}_{n}=\left\{0.8,-4,20,-100,…\right\}[/latex]

38. [latex]{a}_{n}=\left\{-1.25,-5,-20,-80,…\right\}[/latex]

39. [latex]{a}_{n}=\left\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125},…\right\}[/latex]

40. [latex]{a}_{n}=\left\{2,\frac{1}{3},\frac{1}{18},\frac{1}{108},…\right\}[/latex]

41. [latex]{a}_{n}=\left\{3,-1,\frac{1}{3},-\frac{1}{9},…\right\}[/latex]

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

42. Let [latex]{a}_{1}=4[/latex], [latex]{a}_{n}=-3{a}_{n - 1}[/latex]. Find [latex]{a}_{8}[/latex].

43. Let [latex]{a}_{n}=-{\left(-\frac{1}{3}\right)}^{n - 1}[/latex]. Find [latex]{a}_{12}[/latex].

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

44. [latex]{a}_{n}=\left\{-1,3,-9,…,2187\right\}[/latex]

45. [latex]{a}_{n}=\left\{2,1,\frac{1}{2},…,\frac{1}{1024}\right\}[/latex]

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

46.

47.

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

48. [latex]\begin{array}{cc}{a}_{1}=1,& r=\frac{1}{2}\end{array}[/latex]

49. [latex]\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n - 1}\end{array}[/latex]

50. [latex]{a}_{n}=27\cdot {0.3}^{n - 1}[/latex]

51. Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are [latex]200[/latex].

52. Use explicit formulas to give two examples of geometric sequences whose 7^th terms are [latex]1024[/latex].

53. Find the 5^th term of the geometric sequence [latex]\left\{b,4b,16b,…\right\}[/latex].

54. Find the 7^th term of the geometric sequence [latex]\left\{64a\left(-b\right),32a\left(-3b\right),16a\left(-9b\right),…\right\}[/latex].

55. At which term does the sequence [latex]\left\{10,12,14.4,17.28,\text{ }…\right\}[/latex] exceed [latex]100?[/latex]

56. At which term does the sequence [latex]\left\{\frac{1}{2187},\frac{1}{729},\frac{1}{243},\frac{1}{81}\text{ }…\right\}[/latex] begin to have integer values?

57. For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\left(\frac{2}{3}\right)}^{n - 1}[/latex] 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 10^th 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 8^th term.

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