Review Exercises

Introduction to Sequences and Series

Find the first 5 terms of the sequence as well as the 30^th term.

1. $a_n=5n-3$

2. $a_n=-4n+3$

3. $a_n=-10n$

4. $a_n=3n$

5. $a_n={\left(-1\right)}^n{\left(n-2\right)}^2$

6. $a_n=\frac{{\left(-1\right)}^n}{2n-1}$

7. $a_n=\frac{2n+1}{n}$

8. $a_n={\left(-1\right)}^{n+1}\left(n-1\right)$

Find the first 5 terms of the sequence.

9. $a_n=\frac{n{x}^n}{2n+1}$

10. $a_n=\frac{{\left(-1\right)}^{n-1}{x}^{n+2}}{n}$

11. $a_n=2^n{x}^{2n}$

12. $a_n={\left(-3x\right)}^{n-1}$

13. $a_n=a_{n-1}+5$ where $a_1=0$

14. $a_n=4a_{n-1}+1$ where $a_1=-2$

15. $a_n=a_{n-2}-3a_{n-1}$ where $a_1=0$ and $a_2=-3$

16. $a_n=5a_{n-2}-a_{n-1}$ where $a_1=-1$ and $a_2=0$

Find the indicated partial sum.

17. 1, 4, 7, 10, 13,…; $S_5$

18. 3, 1, −1, −3, −5,…; $S_5$

19. −1, 3, −5, 7, −9,…; $S_4$

20. $a_n={\left(-1\right)}^n{n}^2$; $S_4$

21. $a_n=-3{\left(n-2\right)}^2$; $S_4$

22. $a_n={\left(-\frac{1}{5}\right)}^{n-2}$; $S_4$

Evaluate.

23. $${\displaystyle \sum _{k=1}^6\left(1-2k\right)}$$
24. $${\displaystyle \sum _{k=1}^4{\left(-1\right)}^{k}3k^2}$$
25. $${\displaystyle \sum _{n=1}^3\frac{n+1}{n}}$$
26. $${\displaystyle \sum _{n=1}^{7}5{\left(-1\right)}^{n-1}}$$
27. $${\displaystyle \sum _{k=4}^8{\left(1-k\right)}^2}$$
28. $${\displaystyle \sum _{k=-2}^2{\left(\frac{2}{3}\right)}^k}$$

Arithmetic Sequences and Series

Write the first 5 terms of the arithmetic sequence given its first term and common difference. Find a formula for its general term.

29. $a_1=6$; $d=5$

30. $a_1=5$; $d=7$

31. $a_1=5$; $d=-3$

32. $a_1=-\frac{3}{2}$; $d=-\frac{1}{2}$

33. $a_1=-\frac{3}{4}$; $d=-\frac{3}{4}$

34. $a_1=-3.6$; $d=1.2$

35. $a_1=7$; $d=0$

36. $a_1=1$; $d=1$

Given the terms of an arithmetic sequence, find a formula for the general term.

37. 10, 20, 30, 40, 50,…

38. −7, −5, −3, −1, 1,…

39. −2, −5, −8, −11, −14,…

40. $-\frac{1}{3}$, 0, $\frac{1}{3}$, $\frac{2}{3}$, 1,…

41. $a_4=11$ and $a_9=26$

42. $a_5=-5$ and $a_{10}=-15$

43. $a_6=6$ and $a_{24}=15$

44. $a_3=-1.4$ and $a_7=1$

Calculate the indicated sum given the formula for the general term of an arithmetic sequence.

45. $a_n=4n-3$; $S_{60}$

46. $a_n=-2n+9$; $S_{35}$

47. $a_n=\frac{1}{5}n-\frac{1}{2}$; $S_{15}$

48. $a_n=-n+\frac{1}{4}$; $S_{20}$

49. $a_n=1.8n-4.2$; $S_{45}$

50. $a_n=-6.5n+3$; $S_{35}$

Evaluate.

51. $${\displaystyle \sum _{n=1}^{22}\left(7n-5\right)}$$
52. $${\displaystyle \sum _{n=1}^{100}\left(1-4n\right)}$$
53. $${\displaystyle \sum _{n=1}^{35}\left(\frac{2}{3}n\right)}$$
54. $${\displaystyle \sum _{n=1}^{30}\left(-\frac{1}{4}n+1\right)}$$
55. $${\displaystyle \sum _{n=1}^{40}\left(2.3n-1.1\right)}$$
56. $${\displaystyle \sum _{n=1}^{300}n}$$

57. Find the sum of the first 175 positive odd integers.

58. Find the sum of the first 175 positive even integers.

59. Find all arithmetic means between $a_1=\frac{2}{3}$ and $a_5=-\frac{2}{3}$

60. Find all arithmetic means between $a_3=-7$ and $a_7=13.$

61. A 5-year salary contract offers $58,200 for the first year with a $4,200 increase each additional year. Determine the total salary obligation over the 5-year period.

62. The first row of seating in a theater consists of 10 seats. Each successive row consists of four more seats than the previous row. If there are 14 rows, how many total seats are there in the theater?

Geometric Sequences and Series

Write the first 5 terms of the geometric sequence given its first term and common ratio. Find a formula for its general term.

63. $a_1=5$; $r=2$

64. $a_1=3$; $r=-2$

65. $a_1=1$; $r=-\frac{3}{2}$

66. $a_1=-4$; $r=\frac{1}{3}$

67. $a_1=1.2$; $r=0.2$

68. $a_1=-5.4$; $r=-0.1$

Given the terms of a geometric sequence, find a formula for the general term.

69. 4, 40, 400,…

70. −6, −30, −150,…

71. 6, $\frac{9}{2}$, $\frac{27}{8}$,…

72. 1, $\frac{3}{5}$, $\frac{9}{25}$,…

73. $a_4=-4$ and $a_9=128$

74. $a_2=-1$ and $a_5=-64$

75. $a_2=-\frac{5}{2}$ and $a_5=-\frac{625}{16}$

76. $a_3=50$ and $a_6=-\mathrm{6,250}$

77. Find all geometric means between $a_1=-1$ and $a_4=64.$

78. Find all geometric means between $a_3=6$ and $a_6=162.$

Calculate the indicated sum given the formula for the general term of a geometric sequence.

79. $a_n=3{\left(4\right)}^{n-1}$; $S_6$

80. $a_n=-5{\left(3\right)}^{n-1}$; $S_{10}$

81. $a_n=\frac{3}{2}{\left(-2\right)}^n$; $S_{14}$

82. $a_n=\frac{1}{5}{\left(-3\right)}^{n+1}$; $S_{12}$

83. $a_n=8{\left(\frac{1}{2}\right)}^{n+2}$; $S_8$

84. $a_n=\frac{1}{8}{\left(-2\right)}^{n+2}$; $S_{10}$

Evaluate.

85. $${\displaystyle \sum _{n=1}^{10}3{\left(-4\right)}^n}$$
86. $${\displaystyle \sum _{n=1}^9-\frac{3}{5}{\left(-2\right)}^{n-1}}$$
87. $${\displaystyle \sum _{n=1}^{\infty }-3{\left(\frac{2}{3}\right)}^n}$$
88. $${\displaystyle \sum _{n=1}^{\infty }\frac{1}{2}{\left(\frac{4}{5}\right)}^{n+1}}$$
89. $${\displaystyle \sum _{n=1}^{\infty }\frac{1}{2}{\left(-\frac{3}{2}\right)}^n}$$
90. $${\displaystyle \sum _{n=1}^{\infty }\frac{3}{2}{\left(-\frac{1}{2}\right)}^n}$$

91. After the first year of operation, the value of a company van was reported to be $40,000. Because of depreciation, after the second year of operation the van was reported to have a value of $32,000 and then $25,600 after the third year of operation. Write a formula that gives the value of the van after the nth year of operation. Use it to determine the value of the van after 10 years of operation.

92. The number of cells in a culture of bacteria doubles every 6 hours. If 250 cells are initially present, write a sequence that shows the number of cells present after every 6-hour period for one day. Write a formula that gives the number of cells after the nth 6-hour period.

93. A ball bounces back to one-half of the height that it fell from. If dropped from 32 feet, approximate the total distance the ball travels.

94. A structured settlement yields an amount in dollars each year n according to the formula $p_n=\mathrm{12,500}{\left(0.75\right)}^{n-1}.$ What is the total value of a 10-year settlement?

Classify the sequence as arithmetic, geometric, or neither.

95. 4, 9, 14,…

96. 6, 18, 54,…

97. −1, $-\frac{1}{2}$, 0,…

98. 10, 30, 60,…

99. 0, 1, 8,…

100. −1, $\frac{2}{3}$, $-\frac{4}{9}$,…

Evaluate.

101. $${\displaystyle \sum _{n=1}^4{n}^2}$$
102. $${\displaystyle \sum _{n=1}^4{n}^3}$$
103. $${\displaystyle \sum _{n=1}^{32}\left(-4n+5\right)}$$
104. $${\displaystyle \sum _{n=1}^{\infty }-2{\left(\frac{1}{5}\right)}^{n-1}}$$
105. $${\displaystyle \sum _{n=1}^8\frac{1}{3}{\left(-3\right)}^n}$$
106. $${\displaystyle \sum _{n=1}^{46}\left(\frac{1}{4}n-\frac{1}{2}\right)}$$
107. $${\displaystyle \sum _{n=1}^{22}\left(3-n\right)}$$
108. $${\displaystyle \sum _{n=1}^{31}2n}$$
109. $${\displaystyle \sum _{n=1}^{28}3}$$
110. $${\displaystyle \sum _{n=1}^{30}3{\left(-1\right)}^{n-1}}$$
111. $${\displaystyle \sum _{n=1}^{31}3{\left(-1\right)}^{n-1}}$$

Binomial Theorem

Evaluate.

112. $8!$

113. $11!$

114. $\frac{10!}{2!6!}$

115. $\frac{9!3!}{8!}$

116. $\frac{\left(n+3\right)!}{n!}$

117. $\frac{\left(n-2\right)!}{\left(n+1\right)!}$

Calculate the indicated binomial coefficient.

118. $$\left(\begin{array}{l}7\\ 4\end{array}\right)$$
119. $$\left(\begin{array}{l}8\\ 3\end{array}\right)$$
120. $$\left(\begin{array}{c}10\\ 5\end{array}\right)$$
121. $$\left(\begin{array}{l}11\\ 10\end{array}\right)$$
122. $$\left(\begin{array}{c}12\\ 0\end{array}\right)$$
123. $$\left(\begin{array}{l}n+1\\ n-1\end{array}\right)$$
124. $$\left(\begin{array}{c}n\\ n-2\end{array}\right)$$

Expand using the binomial theorem.

125. ${\left(x+7\right)}^3$

126. ${\left(x-9\right)}^3$

127. ${\left(2y-3\right)}^4$

128. ${\left(y+4\right)}^4$

129. ${\left(x+2y\right)}^5$

130. ${\left(3x-y\right)}^5$

131. ${\left(u-v\right)}^6$

132. ${\left(u+v\right)}^6$

133. ${\left(5x^2+2y^2\right)}^4$

134. ${\left(x^3-2y^2\right)}^4$

Answers

1. 2, 7, 12, 17, 22; $a_{30}=147$

3. −10, −20, −30, −40, −50; $a_{30}=-300$

5. −1, 0, −1, 4, −9; $a_{30}=784$

7. 3, $\frac{5}{2}$, $\frac{7}{3}$, $\frac{9}{4}$, $\frac{11}{5}$; $a_{30}=\frac{61}{30}$

9. $\frac{x}{3},\frac{2x^2}{5},\frac{3x^3}{7},\frac{4x^4}{9},\frac{5x^5}{11}$

11. $2x^2,4x^4,8x^6,16x^8,32x^{10}$

13. 0, 5, 10, 15, 20

15. 0, −3, 9, −30, 99

17. 35

19. −5

21. −18

23. −36

25. $\frac{29}{6}$

27. 135

29. 6, 11, 16, 21, 26; $a_n=5n+1$

31. 5, 2, −1, −4, −7; $a_n=8-3n$

33. $-\frac{3}{4}$, $-\frac{3}{2}$, $-\frac{9}{4}$, −3, $-\frac{15}{4}$; $a_n=-\frac{3}{4}n$

35. 7, 7, 7, 7, 7; $a_n=7$

37. $a_n=10n$

39. $a_n=1-3n$

41. $a_n=3n-1$

43. $a_n=\frac{1}{2}n+3$

45. 7,140

47. $\frac{33}{2}$

49. 1,674

51. 1,661

53. 420

55. 1,842

57. 30,625

59. $\frac{1}{3}$, 0, $-\frac{1}{3}$

61. $333,000

63. 5, 10, 20, 40, 80; $a_n=5{\left(2\right)}^{n-1}$

65. 1, $-\frac{3}{2}$, $\frac{9}{4}$, $-\frac{27}{8}$, $\frac{81}{16}$; $a_n={\left(-\frac{3}{2}\right)}^{n-1}$

67. 1.2, 0.24, 0.048, 0.0096, 0.00192; $a_n=1.2{\left(0.2\right)}^{n-1}$

69. $a_n=4{\left(10\right)}^{n-1}$

71. $a_n=6{\left(\frac{3}{4}\right)}^{n-1}$

73. $a_1=\frac{1}{2}{\left(-2\right)}^{n-1}$

75. $a_n=-{\left(\frac{5}{2}\right)}^{n-1}$

77. 4, −16

79. 4,095

81. 16,383

83. $\frac{255}{128}$

85. 2,516,580

87. −6

89. No sum

91. $v_n=\mathrm{40,000}{\left(0.8\right)}^{n-1}$; $v_{10}=\$\mathrm{5,368.71}$

93. 96 feet

95. Arithmetic; $d=5$

97. Arithmetic; $d=\frac{1}{2}$

99. Neither

101. 30

103. −1,952

105. 1,640

107. −187

109. 84

111. 3

113. 39,916,800

115. 54

117. $\frac{1}{n\left(n+1\right)\left(n-1\right)}$

119. 56

121. 11

123. $\frac{n\left(n+1\right)}{2}$

125. $x^3+21x^2+147x+343$

127. $16y^4-96y^3+216y^2-216y+81$

129. $x^5+10x^{4}y+40x^3{y}^2+80x^2{y}^3+80x{y}^4+32y^5$

131. $$\begin{array}{c}{u}^6-6u^{5}v+15u^4{v}^2-20u^3{v}^3\hfill \\ +15u^2{v}^4-6u{v}^5+v^6\hfill \end{array}$$

133. $625x^8+\mathrm{1,000}{x}^6{y}^2+600x^4{y}^4+160x^2{y}^6+16y^8$

Sample Exam

Find the first 5 terms of the sequence.

1. $a_n=6n-15$

2. $a_n=5{\left(-4\right)}^{n-2}$

3. $a_n=\frac{n-1}{2n-1}$

4. $a_n={\left(-1\right)}^{n-1}{x}^{2n}$

Find the indicated partial sum.

5. $a_n=\left(n-1\right)n^2$; $S_4$

6. $${\displaystyle \sum _{k=1}^5{\left(-1\right)}^k{2}^{k-2}}$$

Classify the sequence as arithmetic, geometric, or neither.

7. −1, $-\frac{3}{2}$, −2,…

8. 1, −6, 36,…

9. $\frac{3}{8}$, $-\frac{3}{4}$, $\frac{3}{2}$,…

10. $\frac{1}{2}$, $\frac{1}{4}$, $\frac{2}{9}$,…

Given the terms of an arithmetic sequence, find a formula for the general term.

11. 10, 5, 0, −5, −10,…

12. $a_4=-\frac{1}{2}$ and $a_9=2$

Given the terms of a geometric sequence, find a formula for the general term.

13. $-\frac{1}{8}$, $-\frac{1}{2}$, −2, −8, −32,…

14. $a_3=1$ and $a_8=-32$

Calculate the indicated sum.

15. $a_n=5-n$; $S_{44}$

16. $a_n={\left(-2\right)}^{n+2}$; $S_{12}$

17. $${\displaystyle \sum _{n=1}^{\infty }4{\left(-\frac{1}{2}\right)}^{n-1}}$$
18. $${\displaystyle \sum _{n=1}^{100}\left(2n-\frac{3}{2}\right)}$$

Evaluate.

19. $\frac{14!}{10!6!}$

20. $$\left(\begin{array}{l}9\\ 7\end{array}\right)$$

21. Determine the sum of the first 48 positive odd integers.

22. The first row of seating in a theater consists of 14 seats. Each successive row consists of two more seats than the previous row. If there are 22 rows, how many total seats are there in the theater?

23. A ball bounces back to one-third of the height that it fell from. If dropped from 27 feet, approximate the total distance the ball travels.

Expand using the binomial theorem.

24. ${\left(x-5y\right)}^4$

25. ${\left(3a+b^2\right)}^5$

Answers

1. −9, −3, 3, 9, 15

3. 0, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{4}{9}$

5. 70

7. Arithmetic

9. Geometric

11. $a_n=15-5n$

13. $a_n=-\frac{1}{8}{\left(4\right)}^{n-1}$

15. −770

17. $\frac{8}{3}$

19. $\frac{\mathrm{1,001}}{30}$

21. 2,304

23. 54 feet

25. $$\begin{array}{c}243a^5+405a^4{b}^2+270a^3{b}^4\hfill \\ +90a^2{b}^6+15a{b}^8+b^{10}\hfill \end{array}$$