SECTION 12.1 Sequences 865 4. How much do you need to invest now at 5% per annum compounded monthly so that in 1 year you will have $10,000? (pp. 345–351) 3. If $1000 is invested at 4% per annum compounded semiannually, how much is in the account after 2 years? (pp. 345–351) 5. A(n) is a function whose domain is the set of positive integers. 6. True or False The notation a5 represents the fifth term of a sequence. 7. True or False If ≥ n 2 is an integer, then n n n ! 1 321 ( ) = − ⋅ ⋅ 8. Multiple Choice The sequence = = − a a a 5, 3 n n 1 1 is an example of a(n) sequence. (a) alternating (b) recursive (c) Fibonacci (d) summation 9. The notation ∑ + + + + = = a a a a a n k n k 1 2 3 1 is an example of notation. 10. Multiple Choice k n 1 2 3 k n 1 ∑ = + + + + = = . (a) n! (b) ( ) + n n 1 2 (c) nk (d) ( )( ) + + n n n 1 2 1 6 Concepts and Vocabulary Skill Building In Problems 11–16, evaluate each factorial expression. 11. 10! 12. 9! 13. 9! 6! 14. 12! 10! 15. 3! 7! 4! 16. 5! 8! 3! In Problems 17–28, write down the first five terms of each sequence. 17. { } { } = s n n 18. { } { } = + s n 1 n 2 19. { } { } = + a n n 2 n 20. { } { } = + b n n 2 1 2 n 21. { } { } ( ) = − + c n 1 n n 1 2 22. ( ) { } { } ( ) = − − − d n n 1 2 1 n n 1 23. { } { } = + s 2 3 1 n n n 24. { } ( ) { } = s 4 3 n n 25. { } { } ( ) ( )( ) = − + + t n n 1 1 2 n n 26. { } { } = a n 3 n n 27. { } { } = b n e n n 28. { } { } = c n 2 n n 2 In Problems 29–36, the given pattern continues. Write down the nth term of a sequence { } an suggested by the pattern. 29. 1 2 , 2 3 , 3 4 , 4 5 , … 30. 1 1 2 , 1 2 3 , 1 3 4 , 1 4 5 , … ⋅ ⋅ ⋅ ⋅ 31. … 1, 1 2 , 1 4 , 1 8 , 32. … 2 3 , 4 9 , 8 27 , 16 81 , 33. 1, 1, 1, 1, 1, 1, − − − … 34. … 1, 1 2 , 3, 1 4 , 5, 1 6 , 7, 1 8 , 35. − − − … 1, 2, 3, 4, 5, 6, 36. − − … 2, 4, 6, 8, 10, In Problems 37–50, a sequence is defined recursively. Write down the first five terms. 37. = = + − a a a 2; 3 n n 1 1 38. = = − − a a a 3; 4 n n 1 1 39. = − = + − a a n a 2; n n 1 1 40. = = − − a a n a 1; n n 1 1 41. = = − a a a 5; 2 n n 1 1 42. = = − − a a a 2; n n 1 1 43. = = − a a a n 3; n n 1 1 44. = − = + − a a n a 2; 3 n n 1 1 45. = = = ⋅ − − a a a a a 1; 2; n n n 1 2 1 2 46. =− = = + − − a a a a na 1; 1; n n n 1 2 2 1 47. = = + − a A a a d ; n n 1 1 48. = = ≠ − a A a ra r ; , 0 n n 1 1 49. = = + − a a a 2; 2 n n 1 1 50. = = − a a a 2; 2 n n 1 1 In Problems 51–60, write out each sum. 51. ∑( ) + = k 2 k n 1 52. ∑( ) + = k2 1 k n 1 53. ∑ = k 2 k n 1 2 54. ∑( ) + = k 1 k n 1 2 55. ∑ = 1 3 k n k 0 56. ∑( ) = 3 2 k n k 0 57. ∑ = − + 1 3 k n k 0 1 1 58. ∑( ) + = − k2 1 k n 0 1 59. ∑( ) − = k 1 ln k n k 2 60. ∑( ) − = + 1 2 k n k k 3 1 In Problems 61–70, express each sum using summation notation. 61. + + + + 1 2 3 20 62. + + + + 1 2 3 8 3 3 3 3 63. + + + + + 1 2 2 3 3 4 13 13 1 64. [ ( ) ] + + + + + − 1 3 5 7 2 12 1
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