Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Determine whether each sequence is geometric. If so, find the common ratio.

7. 1, 2, 4, 8, …
8. 1, 2, 3, 4, …
9. 1 , − 2 , 4 , − 8 , … 1 comma negative 2 comma . 4 comma , negative 8 comma dot dot dot
10. − 1 , 1 , − 1 , 1 , … negative 1 comma 1 comma negative 1 comma . 1 comma dot dot dot
11. 10, 4, 1.6, 0.64, …
12. 7, 0.7, 0.07, 0.007, …
13. 18 , − 6 , 2 , − 2 3 , … 18 comma negative 6 comma . 2 comma , negative , 2 thirds , comma dot dot dot
14. 1 , 1 2 , 1 3 , 1 4 , … 1 comma , 1 half , comma , 1 third , comma , 1 fourth , comma dot dot dot
15. 10, 15, 22.5, 33.75, …
16. 2 , − 10 , 50 , − 250 , … 2 comma negative 10 comma . 50 comma . negative 250 comma dot dot dot
17. − 1 , − 6 , − 36 , − 216 , … negative 1 comma negative 6 comma . negative 36 comma . negative 216 comma dot dot dot
18. 1 2 , 1 4 , 1 6 , 1 8 , … 1 half , comma , 1 fourth , comma , 1 sixth , comma , 1 eighth , comma dot dot dot

See Problem 2.

Find the eighth term of each geometric sequence.

19. 3, 9, 27, …
20. − 3 , 6 , − 12 , … negative 3 comma 6 comma negative 12 comma dot dot dot
21. 10, 5, 2.5, …
22. 2 3 , 4 9 , 8 27 , … 2 thirds , comma , 4 ninths , comma , 8 twenty sevenths , comma dot dot dot
23. 24 , − 6 , 3 2 , … 24 comma negative 6 comma . 3 halves , comma dot dot dot
24. − 30 , 7.5 , − 1.875 , … negative 30 comma 7.5 comma negative , 1.875 , comma dot dot dot

See Problem 3.

25. Science When radioactive substances decay, the amount remaining will form a geometric sequence when measured over constant intervals of time. The table shows the amount of Np-240, a radioactive isotope of Neptunium, initially and after 2 hours. What are the amounts left after 1 hour, 3 hours, and 4 hours?

    Hours Elapsed	0	1	2	3	4
    Grams of Np-240	1244	□ white square	346	□ white square	□ white square

See Problem 4.

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.

26. 5 , □ , 911.25 , … 5 comma white square comma , 911.25 , comma dot dot dot
27. 9180 , □ , 255 , … 9180 , comma white square comma 255 comma dot dot dot
28. 2 5 , □ , 8 45 , … 2 fifths , comma white square comma , 8 forty fifths , comma dot dot dot
29. 3 , □ , 0.75 , … 3 comma white square comma , 0.75 , comma dot dot dot
30. 5 , □ , 2.8125 , … 5 comma white square comma , 2.8125 , comma dot dot dot
31. 12 , □ , 3 , … 12 comma white square comma 3 comma dot dot dot

B Apply

Write an explicit formula for each sequence. Then generate the first five terms.

32. a 1 = 1 , r = 0.5 eh sub 1 , equals 1 . comma , r equals 0.5
33. a 1 = 100 , r = − 20 eh sub 1 , equals 100 . comma r equals negative 20
34. a 1 = 7 , r = 1 eh sub 1 , equals 7 . comma , r equals 1
35. a 1 = 1024 , r = 0.5 eh sub 1 , equals , 1024 . comma , r equals 0.5
36. a 1 = 4 , r = 0.1 eh sub 1 , equals 4 . comma , r equals 0.1
37. a 1 = 10 , r = − 1 eh sub 1 , equals 10 . comma r equals negative 1

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.

38. 45, 90, 180, 360, …
39. 25, 50, 75, 100, …
40. 3 , − 3 , 3 , − 3 , … 3 comma negative 3 comma . 3 comma , negative 3 comma dot dot dot
41. − 5 , 10 , − 20 , 40 , … negative 5 comma 10 comma negative 20 comma . 40 comma dot dot dot
42. 2, 1, 0.5, 0.25, …
43. 1, 4, 9, 16, …

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.)

44. 972 , □ , □ , □ , 12 , … 972 comma white square comma white square comma white square comma 12 comma dot dot dot
45. 2.5 , □ , □ , □ , 202.5 , … 2.5 comma white square comma white square comma white square comma , 202.5 , comma dot dot dot
46. 12.5 , □ , □ , □ , 5.12 , … 12.5 , comma white square comma white square comma white square comma , 5.12 , comma dot dot dot
47. − 4 , □ , □ , □ , − 30 3 8 , … negative 4 comma white square comma white square comma white square comma negative 30 , and 3 eighths , comma dot dot dot

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