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Solutions for Arithmetic Sequences

Solutions to Try Its

1. The sequence is arithmetic. The common difference is −2-2. 2. The sequence is not arithmetic because 3−1≠6−33 - 1\ne 6 - 3. 3. {1,6,11,16,21}\left\{1, 6, 11, 16, 21\right\} 4. a2=2{a}_{2}=2 5. a1=25an=an−1+12, for n≥2\begin{array}{l}{a}_{1}=25\hfill \\ {a}_{n}={a}_{n - 1}+12,\text{ for }n\ge 2\hfill \end{array} 6. an=53−3n{a}_{n}=53 - 3n 7. There are 11 terms in the sequence. 8. The formula is Tn=10+4n{T}_{n}=10+4n, 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 12\frac{1}{2} 9. The sequence is not arithmetic because 16−4≠64−1616 - 4\ne 64 - 16. 11. 0,23,43,2,830,\frac{2}{3},\frac{4}{3},2,\frac{8}{3} 13. 0,−5,−10,−15,−200,-5,-10,-15,-20 15. a4=19{a}_{4}=19 17. a6=41{a}_{6}=41 19. a1=2{a}_{1}=2 21. a1=5{a}_{1}=5 23. a1=6{a}_{1}=6 25. a21=−13.5{a}_{21}=-13.5 27. −19,−20.4,−21.8,−23.2,−24.6-19,-20.4,-21.8,-23.2,-24.6 29. a1=17;an=an−1+9n≥2\begin{array}{ll}{a}_{1}=17; {a}_{n}={a}_{n - 1}+9\hfill & n\ge 2\hfill \end{array} 31. a1=12;an=an−1+5n≥2\begin{array}{ll}{a}_{1}=12; {a}_{n}={a}_{n - 1}+5\hfill & n\ge 2\hfill \end{array} 33. a1=8.9;an=an−1+1.4n≥2\begin{array}{ll}{a}_{1}=8.9; {a}_{n}={a}_{n - 1}+1.4\hfill & n\ge 2\hfill \end{array} 35. a1=15;an=an−1+14n≥2\begin{array}{ll}{a}_{1}=\frac{1}{5}; {a}_{n}={a}_{n - 1}+\frac{1}{4}\hfill & n\ge 2\hfill \end{array} 37. 1=16;an=an−1−1312n≥2\begin{array}{ll}{}_{1}=\frac{1}{6}; {a}_{n}={a}_{n - 1}-\frac{13}{12}\hfill & n\ge 2\hfill \end{array} 39. a1=4; an=an−1+7; a14=95{a}_{1}=4;\text{ }{a}_{n}={a}_{n - 1}+7;\text{ }{a}_{14}=95 41. First five terms: 20,16,12,8,420,16,12,8,4. 43. an=1+2n{a}_{n}=1+2n 45. an=−105+100n{a}_{n}=-105+100n 47. an=1.8n{a}_{n}=1.8n 49. an=13.1+2.7n{a}_{n}=13.1+2.7n 51. an=13n−13{a}_{n}=\frac{1}{3}n-\frac{1}{3} 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. 1,4,7,10,13,16,191,4,7,10,13,16,19 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: an=20.6n{a}_{n}=20.6n and an=2+20.4n.{a}_{n}=2+20.4\mathrm{n.} 69. a11=−17a+38b{a}_{11}=-17a+38b 71. The sequence begins to have negative values at the 13th term, a13=−13{a}_{13}=-\frac{1}{3} 73. Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a1=3,an=an−1−3{a}_{1}=3,{a}_{n}={a}_{n - 1}-3. First 4 terms: 3,0,−3,−6a31=−87\begin{array}{ll}3,0,-3,-6\hfill & {a}_{31}=-87\hfill \end{array}

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