Solutions for Arithmetic Sequences

Solutions to Try Its

1. The sequence is arithmetic. The common difference is [latex]-2[/latex].

2. The sequence is not arithmetic because [latex]3 - 1\ne 6 - 3[/latex].

3. [latex]\left\{1, 6, 11, 16, 21\right\}[/latex]

4. [latex]{a}_{2}=2[/latex]

5. [latex]\begin{array}{l}{a}_{1}=25\hfill \\ {a}_{n}={a}_{n - 1}+12,\text{ for }n\ge 2\hfill \end{array}[/latex]

6. [latex]{a}_{n}=53 - 3n[/latex]

7. There are 11 terms in the sequence.

8. The formula is [latex]{T}_{n}=10+4n[/latex], and it will take her 42 minutes.

Solutions to Odd-Numbered Exercises

1. A sequence where each successive term of the sequence increases (or decreases) by a constant value.

3. We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

5. Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

7. The common difference is [latex]\frac{1}{2}[/latex]

9. The sequence is not arithmetic because [latex]16 - 4\ne 64 - 16[/latex].

11. [latex]0,\frac{2}{3},\frac{4}{3},2,\frac{8}{3}[/latex]

13. [latex]0,-5,-10,-15,-20[/latex]

15. [latex]{a}_{4}=19[/latex]

17. [latex]{a}_{6}=41[/latex]

19. [latex]{a}_{1}=2[/latex]

21. [latex]{a}_{1}=5[/latex]

23. [latex]{a}_{1}=6[/latex]

25. [latex]{a}_{21}=-13.5[/latex]

27. [latex]-19,-20.4,-21.8,-23.2,-24.6[/latex]

29. [latex]\begin{array}{ll}{a}_{1}=17; {a}_{n}={a}_{n - 1}+9\hfill & n\ge 2\hfill \end{array}[/latex]

31. [latex]\begin{array}{ll}{a}_{1}=12; {a}_{n}={a}_{n - 1}+5\hfill & n\ge 2\hfill \end{array}[/latex]

33. [latex]\begin{array}{ll}{a}_{1}=8.9; {a}_{n}={a}_{n - 1}+1.4\hfill & n\ge 2\hfill \end{array}[/latex]

35. [latex]\begin{array}{ll}{a}_{1}=\frac{1}{5}; {a}_{n}={a}_{n - 1}+\frac{1}{4}\hfill & n\ge 2\hfill \end{array}[/latex]

37. [latex]\begin{array}{ll}{}_{1}=\frac{1}{6}; {a}_{n}={a}_{n - 1}-\frac{13}{12}\hfill & n\ge 2\hfill \end{array}[/latex]

39. [latex]{a}_{1}=4;\text{ }{a}_{n}={a}_{n - 1}+7;\text{ }{a}_{14}=95[/latex]

41. First five terms: [latex]20,16,12,8,4[/latex].

43. [latex]{a}_{n}=1+2n[/latex]

45. [latex]{a}_{n}=-105+100n[/latex]

47. [latex]{a}_{n}=1.8n[/latex]

49. [latex]{a}_{n}=13.1+2.7n[/latex]

51. [latex]{a}_{n}=\frac{1}{3}n-\frac{1}{3}[/latex]

53. There are 10 terms in the sequence.

55. There are 6 terms in the sequence.

57. The graph does not represent an arithmetic sequence.

59.
Graph of a scattered plot with labeled points: (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31). The x-axis is labeled n and the y-axis is labeled a_n.

61. [latex]1,4,7,10,13,16,19[/latex]

63.
Graph of a scattered plot with labeled points: (1, 1), (2, 4), (3, 7), (4, 10), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n.

65.
Graph of a scattered plot with labeled points: (1, 5.5), (2, 6), (3, 6.5), (4, 7), and (5, 7.5). The x-axis is labeled n and the y-axis is labeled a_n.

67. Answers will vary. Examples: [latex]{a}_{n}=20.6n[/latex] and [latex]{a}_{n}=2+20.4\mathrm{n.}[/latex]

69. [latex]{a}_{11}=-17a+38b[/latex]

71. The sequence begins to have negative values at the 13th term, [latex]{a}_{13}=-\frac{1}{3}[/latex]

73. Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: [latex]{a}_{1}=3,{a}_{n}={a}_{n - 1}-3[/latex]. First 4 terms: [latex]\begin{array}{ll}3,0,-3,-6\hfill & {a}_{31}=-87\hfill \end{array}[/latex]