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1 சே (-1) 201 27 4. a. - 2 + (-1)

1 சே (-1) 201 27 4. a. - 2 + (-1)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 81E
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Plis nit 18 nd sd 14 m 1458 Exercises 10.1. Pinding Terms of a Sequence Each of Exercises 1-6 gives a formula for the nth term a, of a se quence (a.). Find the values of a, a, ay, and La. - S. 4. m (-1)^*¹ 2n-1 2 I 17. Each of Exercises 7-12 gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first terms of the sequence. 7.-1, += + (1/2") 8. a 1, 0+1=a₂/(+1) 9. ₁2. - (-1)/2 10.-2, ₂-1 / (+1) +24+Or 11. aa1₁ 12. ₁2. =-1₁ = ₂1/an 4.2+(-1)" 6. 0 = Finding a Sequence's Formula In Exercises 13-26, find a formula for the nth term of the sequence 13. The sequence 1,-1,1.-1, 1.. with alternating signs 14. The sequence-1.1.-1.1.-1.. 15. The sequence 1.-4, 9, -16, 25... with alternating signs Squares of the positive integers, with alternating signs Reciprocals of squares of the positive integers with alternating sites Powers of 1 divided by multiples of 3 16. The sequence 1.-1625 49 12 2² 23 24 9 12 15 18 21 23. 1135 6 12 20 30 18.-2-12 10.1 Sequences 559 It is important to realize that Theorem 6 does not say that convergent sequences are monotonic. The sequence ((-1)/n) converges and is bounded, but it is not monotonic since it alternates between positive and negative values as it tends toward zero. What the theorem does say is that a nondecreasing sequence converges when it is bounded from above, but it diverges to infinity otherwise. 19. The sequence 0, 3, 8, 15, 24,... 20. The sequence -3, -2,-1.0, 1..... 21. The sequence 1, 5, 9, 13, 17,... 22. The sequence 2, 6, 10, 14, 18,... 58 11 14 17 12 6 24 120 Integers differing by 2 divided by products of consecutive imegers Squares of the positive integers diminished by 1 Integers, beginning with-3 Every other odd positive integer Every other even positive integer Integers differing by 3 divided by factorials. 64 25 125 625 3125 15,625 24. 25. The sequence 1.0, 1.0, 1.... 26. The sequence 0, 1, 1, 2, 2, 3, 3, 4,... Convergence and Divergence Which of the sequences (a.) in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence. n+ (-1) 27. 2+ (0.1) 1-2N 1 + 2n 0₁- 31. d. 33. G 37. 1-3 +8n³ m² - 2 + 1 8-1 35. 01 (-1)^ = (^2-¹)(¹-²) (-1)^² 39.-1 2n Va+1 41. . 45. 4, = 49. a. - SIG A In(n+1) Vn 51. 4. - 8 (₁ + ²)* 53. a. = 55.-10 57. " 59. 4,- In a AU/ 1/4 28. - 30. 32.d.- Cubes of positive integers divided by powers of 5 42. a 40. 4. - Alternating I's and 0% Each positive integer repeated 34. a.- 1- 70-46² 36. 2. - (-11 (1-1) 38. 4. = (2-4) (+) 46. a. = 48. 4. 50. a. (0.91" 44. 2. cos (A) n 2+1 1-3√₂ ²5n+6 52a,- sinn Inn in 2 (0.03) (1-4) 54. a.- 56. a. - V 58., (+4)* 60. a.Inn-In (n + 1)
Transcribed Image Text:Plis nit 18 nd sd 14 m 1458 Exercises 10.1. Pinding Terms of a Sequence Each of Exercises 1-6 gives a formula for the nth term a, of a se quence (a.). Find the values of a, a, ay, and La. - S. 4. m (-1)^*¹ 2n-1 2 I 17. Each of Exercises 7-12 gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first terms of the sequence. 7.-1, += + (1/2") 8. a 1, 0+1=a₂/(+1) 9. ₁2. - (-1)/2 10.-2, ₂-1 / (+1) +24+Or 11. aa1₁ 12. ₁2. =-1₁ = ₂1/an 4.2+(-1)" 6. 0 = Finding a Sequence's Formula In Exercises 13-26, find a formula for the nth term of the sequence 13. The sequence 1,-1,1.-1, 1.. with alternating signs 14. The sequence-1.1.-1.1.-1.. 15. The sequence 1.-4, 9, -16, 25... with alternating signs Squares of the positive integers, with alternating signs Reciprocals of squares of the positive integers with alternating sites Powers of 1 divided by multiples of 3 16. The sequence 1.-1625 49 12 2² 23 24 9 12 15 18 21 23. 1135 6 12 20 30 18.-2-12 10.1 Sequences 559 It is important to realize that Theorem 6 does not say that convergent sequences are monotonic. The sequence ((-1)/n) converges and is bounded, but it is not monotonic since it alternates between positive and negative values as it tends toward zero. What the theorem does say is that a nondecreasing sequence converges when it is bounded from above, but it diverges to infinity otherwise. 19. The sequence 0, 3, 8, 15, 24,... 20. The sequence -3, -2,-1.0, 1..... 21. The sequence 1, 5, 9, 13, 17,... 22. The sequence 2, 6, 10, 14, 18,... 58 11 14 17 12 6 24 120 Integers differing by 2 divided by products of consecutive imegers Squares of the positive integers diminished by 1 Integers, beginning with-3 Every other odd positive integer Every other even positive integer Integers differing by 3 divided by factorials. 64 25 125 625 3125 15,625 24. 25. The sequence 1.0, 1.0, 1.... 26. The sequence 0, 1, 1, 2, 2, 3, 3, 4,... Convergence and Divergence Which of the sequences (a.) in Exercises 27-90 converge, and which diverge? Find the limit of each convergent sequence. n+ (-1) 27. 2+ (0.1) 1-2N 1 + 2n 0₁- 31. d. 33. G 37. 1-3 +8n³ m² - 2 + 1 8-1 35. 01 (-1)^ = (^2-¹)(¹-²) (-1)^² 39.-1 2n Va+1 41. . 45. 4, = 49. a. - SIG A In(n+1) Vn 51. 4. - 8 (₁ + ²)* 53. a. = 55.-10 57. " 59. 4,- In a AU/ 1/4 28. - 30. 32.d.- Cubes of positive integers divided by powers of 5 42. a 40. 4. - Alternating I's and 0% Each positive integer repeated 34. a.- 1- 70-46² 36. 2. - (-11 (1-1) 38. 4. = (2-4) (+) 46. a. = 48. 4. 50. a. (0.91" 44. 2. cos (A) n 2+1 1-3√₂ ²5n+6 52a,- sinn Inn in 2 (0.03) (1-4) 54. a.- 56. a. - V 58., (+4)* 60. a.Inn-In (n + 1)
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