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.
61. [latex]1,4,7,10,13,16,19[/latex]
63.
65.
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]