Sequences and Their Notations
7. First four terms: [latex]-8,\text{ }-\frac{16}{3},\text{ }-4,\text{ }-\frac{16}{5}[/latex]
11. First four terms: [latex]1.25,\text{ }-5,\text{ }20,\text{ }-80[/latex] .
13. First four terms: [latex]\frac{1}{3},\text{ }\frac{4}{5},\text{ }\frac{9}{7},\text{ }\frac{16}{9}[/latex] .
21. [latex]{a}_{n}={n}^{2}+3[/latex]
23. [latex]{a}_{n}=\frac{{2}^{n}}{2n}\text{ or }\frac{{2}^{n - 1}}{n}[/latex]
25. [latex]{a}_{n}={\left(-\frac{1}{2}\right)}^{n - 1}[/latex]
27. First five terms: [latex]3,\text{ }-9,\text{ }27,\text{ }-81,\text{ }243[/latex]
29. First five terms: [latex]-1,\text{ }1,\text{ }-9,\text{ }\frac{27}{11},\text{ }\frac{891}{5}[/latex]
35. [latex]{a}_{1}=-8,{a}_{n}={a}_{n - 1}+n[/latex]
37. [latex]{a}_{1}=35,{a}_{n}={a}_{n - 1}+3[/latex]
43. First four terms: [latex]1,\frac{1}{2},\frac{2}{3},\frac{3}{2}[/latex]
45. First four terms: [latex]-1,2,\frac{6}{5},\frac{24}{11}[/latex]
55. [latex]{a}_{1}=6,\text{ }{a}_{n}=2{a}_{n - 1}-5[/latex]
67. If [latex]{a}_{n}=-421[/latex] is a term in the sequence, then solving the equation [latex]-421=-6 - 8n[/latex] for [latex]n[/latex] will yield a non-negative integer. However, if [latex]-421=-6 - 8n[/latex], then [latex]n=51.875[/latex] so [latex]{a}_{n}=-421[/latex] is not a term in the sequence.
Arithmetic Sequences
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.
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]
27. [latex]-19,-20.4,-21.8,-23.2,-24.6[/latex]
41. First five terms: [latex]20,16,12,8,4[/latex]
29. [latex]\begin{array}{ll}{a}_{1}=17; {a}_{n}={a}_{n - 1}+9\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]
45. [latex]{a}_{n}=-105+100n[/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.
Geometric Sequences
1. A sequence in which the ratio between any two consecutive terms is constant.
3. Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.
7. The common ratio is [latex]-2[/latex]
9. The sequence is geometric. The common ratio is 2.
11. The sequence is geometric. The common ratio is [latex]-\frac{1}{2}[/latex].
13. The sequence is geometric. The common ratio is [latex]5[/latex].
15. [latex]5,1,\frac{1}{5},\frac{1}{25},\frac{1}{125}[/latex]
17. [latex]800,400,200,100,50[/latex]
23. [latex]7,1.4,0.28,0.056,0.0112[/latex]
33. [latex]12,-6,3,-\frac{3}{2},\frac{3}{4}[/latex]
19. [latex]{a}_{4}=-\frac{16}{27}[/latex]
21. [latex]{a}_{7}=-\frac{2}{729}[/latex]
43. [latex]{a}_{12}=\frac{1}{177,147}[/latex]
25. [latex]\begin{array}{cc}a{}_{1}=-32,& {a}_{n}=\frac{1}{2}{a}_{n - 1}\end{array}[/latex]
27. [latex]\begin{array}{cc}{a}_{1}=10,& {a}_{n}=-0.3{a}_{n - 1}\end{array}[/latex]
29. [latex]\begin{array}{cc}{a}_{1}=\frac{3}{5},& {a}_{n}=\frac{1}{6}{a}_{n - 1}\end{array}[/latex]
31. [latex]{a}_{1}=\frac{1}{512},{a}_{n}=-4{a}_{n - 1}[/latex]
35. [latex]{a}_{n}={3}^{n - 1}[/latex]
37. [latex]{a}_{n}=0.8\cdot {\left(-5\right)}^{n - 1}[/latex]
39. [latex]{a}_{n}=-{\left(\frac{4}{5}\right)}^{n - 1}[/latex]
41. [latex]{a}_{n}=3\cdot {\left(-\frac{1}{3}\right)}^{n - 1}[/latex]
45. There are [latex]12[/latex] terms in the sequence.
51. Answers will vary. Examples: [latex]{\begin{array}{cc}{a}_{1}=800,& {a}_{n}=0.5a\end{array}}_{n - 1}[/latex] and [latex]{\begin{array}{cc}{a}_{1}=12.5,& {a}_{n}=4a\end{array}}_{n - 1}[/latex]
53. [latex]{a}_{5}=256b[/latex]
Series and Their Notations
1. An [latex]n\text{th}[/latex] partial sum is the sum of the first [latex]n[/latex] terms of a sequence.
3. A geometric series is the sum of the terms in a geometric sequence.
7. [latex]\sum _{n=0}^{4}5n[/latex]
9. [latex]\sum _{k=1}^{5}4[/latex]
11. [latex]\sum _{k=1}^{20}8k+2[/latex]
17. [latex]\sum _{k=1}^{7}8\cdot {0.5}^{k - 1}[/latex]
13. [latex]{S}_{5}=\frac{5\left(\frac{3}{2}+\frac{7}{2}\right)}{2}[/latex]
15. [latex]{S}_{13}=\frac{13\left(3.2+5.6\right)}{2}[/latex]
35. [latex]{S}_{7}=\frac{147}{2}[/latex]
37. [latex]{S}_{11}=\frac{55}{2}[/latex]
19. [latex]{S}_{5}=\frac{9\left(1-{\left(\frac{1}{3}\right)}^{5}\right)}{1-\frac{1}{3}}=\frac{121}{9}\approx 13.44[/latex]
21. [latex]{S}_{11}=\frac{64\left(1-{0.2}^{11}\right)}{1 - 0.2}=\frac{781,249,984}{9,765,625}\approx 80[/latex]
39. [latex]{S}_{7}=5208.4[/latex]
41. [latex]{S}_{10}=-\frac{1023}{256}[/latex]
23. The series is defined. [latex]S=\frac{2}{1 - 0.8}[/latex]
25. The series is defined. [latex]S=\frac{-1}{1-\left(-\frac{1}{2}\right)}[/latex]
31. 49
33. 254
43. [latex]S=-\frac{4}{3}[/latex]
45. [latex]S=9.2[/latex]
47. $3,705.42
49. $695,823.97
57. $400 per month
59. 420 feet
61. 12 feet