Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Find the first six terms of each sequence.

7. a n = 3 n + 2 eh sub n , equals , 3 n plus 2
8. a n = − 5 n + 1 eh sub n , equals negative 5 n plus 1
9. a n = 1 2 n eh sub n , equals , 1 half , n
10. a n = n 2 + 1 eh sub n , equals , n squared , plus 1
11. a n = 3 n 2 − n eh sub n , equals . 3 n squared , minus n
12. a n = 2 n − 1 eh sub n , equals , 2 to the n , minus 1
13. a n = 1 2 n 3 − 1 eh sub n , equals , 1 half , n cubed , minus 1
14. a n = ( − 3 ) n eh sub n , equals . open negative 3 close to the n

See Problem 2.

Write a recursive definition for each sequence.

15. 80, 77, 74, 71, 68, …
16. 4, 8, 16, 32, 64, …
17. 0, 3, 7, 12, 18, …
18. 1, 4, 7, 10, 13, …
19. 100, 10, 1, 0.1, 0.01, …
20. 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , … 1 half , comma , 1 fourth , comma , 1 eighth , comma , 1 sixteenth , comma , 1 thirty second , comma dot dot dot
21. 4 , − 8 , 16 , − 32 , 64 , … 4 comma negative 8 comma 16 comma negative 32 comma 64 comma dot dot dot
22. 1 , 2 , 6 , 24 , 120 , … 1 comma 2 comma 6 comma 24 comma 120 comma dot dot dot
23. 1, 5, 14, 30, …

See Problem 3.

Write an explicit formula for each sequence. Find the tenth term.

24. 4, 5, 6, 7, 8, …
25. 4, 7, 10, 13, 16, …
26. 3, 7, 11, 15, 19, …
27. − 2 1 2 , − 2 , − 1 1 2 , − 1 , … negative 2 , and 1 half , comma negative 2 comma negative 1 , and 1 half , comma negative 1 comma dot dot dot
28. 1, 4, 9, 16, …
29. 2, 5, 10, 17, 26, …
30. 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , … 1 half , comma , 1 third , comma , 1 fourth , comma , 1 fifth , comma , 1 sixth , comma dot dot dot
31. 1, 3, 9, 27, …
32. 1 2 , − 1 4 , 1 8 , − 1 16 , … 1 half , comma negative , 1 fourth , comma , 1 eighth , comma negative , 1 sixteenth , comma dot dot dot

Find the eighth term of each sequence.

33. − 2 , − 1 , 0 , 1 , 2 , … negative 2 comma negative 1 comma 0 comma 1 comma 2 comma dot dot dot
34. 43, 41, 39, 37, 35, …
35. 40 , 20 , 10 , 5 , 5 2 , … 40 comma 20 comma 10 comma 5 comma , 5 halves , comma dot dot dot
36. 6 , 1 , − 4 , − 9 , … 6 comma 1 comma negative 4 comma negative 9 comma dot dot dot
37. 144 , 36 , 9 , 9 4 , … 144 comma 36 comma 9 comma , 9 fourths , comma dot dot dot
38. 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , … 1 half , comma , 1 fourth , comma , 1 eighth , comma , 1 sixteenth , comma , 1 thirty second , comma dot dot dot
39. 2 , 1 , − 2 , − 7 , − 14 , … 2 comma 1 comma negative 2 comma negative 7 comma . negative 14 comma dot dot dot
40. 3 4 , − 3 2 , 3 , − 6 , … 3 fourths , comma negative , 3 halves , comma 3 comma negative 6 comma dot dot dot
41. 2 , − 3 2 , 4 3 , − 5 4 , … 2 comma negative , 3 halves , comma , 4 thirds , comma negative , 5 fourths , comma dot dot dot

See Problem 4.

42. Exercise You walk 1 mile the first day of your training, 1.2 miles the second day, 1.6 miles the third day, and 2.4 miles the fourth day. If you continue this pattern, how many miles do you walk the seventh day?

B Apply

Determine whether each formula is explicit or recursive. Then find the first five terms of each sequence.

43. a n = 2 a n − 1 + 3 , eh sub n , equals . 2 eh sub n minus 1 end sub . plus 3 comma  where a 1 = 3 eh sub 1 , equals 3
44. a n = 1 2 ( n ) ( n − 1 ) eh sub n , equals , 1 half . open n close . open , n minus 1 , close
45. a n = ( n − 5 ) ( n + 5 ) eh sub n , equals open n minus 5 close open n plus 5 close
46. a n = − 3 a n − 1 , eh sub n , equals . negative 3 eh sub n minus 1 end sub . comma  where a 1 = − 2 eh sub 1 , equals negative 2
47. a n = − 4 n 2 − 2 eh sub n , equals negative , 4 n squared , minus 2
48. a n = 2 n 2 + 1 eh sub n , equals . 2 n squared , plus 1

Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence.

49. a n = ( n + 1 ) 2 eh sub n , equals . open n plus 1 close squared
50. a n = 2 ( n − 1 ) 3 eh sub n , equals . 2 open n minus 1 close cubed
51. a n = n 2 n + 1 eh sub n , equals . fraction n squared , over n plus 1 end fraction
52. a n = n + 1 n + 2 eh sub n , equals . fraction n plus 1 , over n plus 2 end fraction

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