Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Evaluate the sum of the finite geometric series.

8. 1 + 2 + 4 + 8 + ⋯ + 128 1 plus 2 plus 4 plus 8 plus math axis ellipsis plus 128
9. 4 + 12 + 36 + 108 + ⋯ + 972 4 plus 12 plus 36 plus 108 plus math axis ellipsis plus 972
10. 3 + 6 + 12 + 24 + 48 + ⋯ + 768 3 plus 6 plus 12 . plus . 24 plus 48 plus math axis ellipsis plus 768
11. − 5 − 10 − 20 − 40 − ⋯ − 2560 negative 5 minus 10 minus 20 minus 40 . minus math axis ellipsis negative , 2560
12. ∑ n = 1 5 3 n sum , from , n equals 1 , to , 5 , of . 3 to the n
13. ∑ n = 1 4 ( 1 2 ) n + 1 sum , from , n equals 1 , to , 4 , of . open , 1 half , close super n plus 1 end super
14. ∑ n = 1 4 ( 2 3 ) n − 1 sum , from , n equals 1 , to , 4 , of . open , 2 thirds , close super n minus 1 end super
15. ∑ n = 1 5 ( 1 3 ) n − 1 sum , from , n equals 1 , to , 5 , of . open , 1 third , close super n minus 1 end super

See Problem 2.

16. Financial Planning In March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost $1375. They start with $125. Each month they plan to deposit 20% more than the previous month. Will they have enough money for their trip? If not, how much more do they need?

See Problem 3.

Determine whether each infinite geometric series diverges or converges. If the series converges, state the sum.

17. 1 + 1 4 + 1 16 + … 1 plus , 1 fourth , plus , 1 sixteenth , plus dot dot dot
18. 1 − 1 2 + 1 4 − … 1 minus , 1 half , plus , 1 fourth , minus dot dot dot
19. 4 + 2 + 1 + … 4 plus 2 plus 1 plus . dot dot dot
20. 1 + 2 + 4 + … 1 plus 2 plus 4 plus . dot dot dot
21. 6 + 18 + 54 + … 6 plus 18 plus 54 plus . dot dot dot
22. − 54 − 18 − 6 − … negative 54 minus 18 minus 6 minus . dot dot dot
23. 1 − 1 + 1 − … 1 minus 1 plus 1 . minus dot dot dot
24. 1 + 1 5 + 1 25 + … 1 plus , 1 fifth , plus , 1 twenty fifth , plus dot dot dot
25. 1 4 + 1 2 + 1 + 2 + … 1 fourth , plus , 1 half , plus 1 plus 2 plus dot dot dot

Evaluate each infinite geometric series.

26. 1.1 + 0.11 + 0.011 + … 1.1 plus , 0.11 , plus , 0.011 , plus dot dot dot
27. 1.1 − 0.11 + 0.011 − … 1.1 minus , 0.11 , plus , 0.011 , minus dot dot dot
28. 1 − 1 5 + 1 25 − 1 125 + … 1 minus , 1 fifth , plus , 1 twenty fifth , minus , 1 over 125 , plus dot dot dot
29. 3 + 1 + 1 3 + 1 9 + … 3 plus 1 plus , 1 third , plus , 1 ninth , plus dot dot dot
30. 3 + 2 + 4 3 + 8 9 + … 3 plus 2 plus , 4 thirds , plus , 8 ninths , plus dot dot dot
31. 3 − 2 + 4 3 − 8 9 + … 3 minus 2 plus , 4 thirds , minus , 8 ninths , plus dot dot dot

B Apply

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms.

32. 2 + 4 + 8 + 16 + … ; n = 10 2 plus 4 plus 8 plus 16 plus . dot dot dot semicolon , n equals 10
33. 2 + 4 + 6 + 8 + … ; n = 20 2 plus 4 plus 6 plus 8 plus . dot dot dot semicolon , n equals 20
34. − 5 + 25 − 125 + 625 − … ; n = 9 negative 5 plus 25 minus 125 . plus 625 minus dot dot dot semicolon , n equals 9
35. 6.4 + 8 + 10 + 12 . 5 + … ; n = 7 6.4 plus 8 plus 10 plus . 12 . 5 plus dot dot dot semicolon , n equals 7
36. 1 + 2 + 3 + 4 + … ; n = 1000 1 plus 2 plus 3 plus 4 plus . dot dot dot semicolon . n equals , 1000
37. 81 + 27 + 9 + 3 + … ; n = 200 81 plus 27 plus 9 plus 3 plus . dot dot dot semicolon , n equals 200
38. Think About a Plan The height a ball bounces is less than the height of the previous bounce due to friction. The heights of the bounces form a geometric sequence. Suppose a ball is dropped from one meter and rebounds to 95% of the height of the previous bounce. What is the total distance traveled by the ball when it comes to rest?
    • Does the problem give you enough information to solve the problem?
    • How can you write the general term of the sequence?
    • What formula should you use to calculate the total distance?

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