Sequences and Series
Arithmetic Sequences
Class Work
Find the common difference in sequence, and then write the next 3 terms in the sequence.
1) 3, 7 ,11, 15, …
2) 1, 8, 15, 22, …
3) 5, 2, -1, -4, …
4) 68, 56, 44, 32, …
5) 1.3, 2.6, 3.9, 5.2, …
Use the given information to write the first 5 terms of the sequence and the 20 th term.
6) a1= 4, d= 8
7) a1= 10, d= -6
8) a2= 9, d= 6
9) a15= 32 , d= 5
10) 4, 7, 10, 13, …
Write the recursive formula for the sequence.
11) 3, 7 ,11, 15, …
12) 5, 2, -1, -4, …
13) 1.3, 2.6, 3.9, 5.2, …
14) a2= 9, d= 6
15) a15= 32 , d= 5
Write the explicit formula for the sequence.
16) a1= 4, d= 8
17) a1= 10, d= -6
18) a2= 9, d= 6
19) 1, 8, 15, 22, …
20) 68, 56, 44, 32, …
Solve the following questions.
21) What is the common difference when a1=8 and a11=40?
22) What is 20th term if a1=12 and a10= -33?
23) If a1=12 , an=111 ,and d = 9, what is n?
24) Tom works at a car dealership selling cars. If he makes $4000 a month and $250 per car he sells, how much did he
make in July if he sold 12 cars? In August he made $6500, how many cars did he sell?
Find the missing term(s) in the sequence.
25) 3, _ , 9
26) 4, _ , _ , 19
27) 5, _ , _ , -7
28) 20, _ , _ , _, 68
Homework
Find the common difference in sequence, and then write the next 3 terms in the sequence.
29) 2, 11, 20, 29, …
30) -17, -6, 5, 16, …
31) 12, 9, 6, 3, …
32) 45, 31, 17, 3, …
33) 4.3, 5.7, 7.1, 8.5, …
Use the given information to write the first 5 terms of the sequence and the 20 th term.
34) a1= 4, d= 3
35) a1= 10, d= -8
36) a2= 9, d= 14
Alg II- Sequences & Series
~1~
NJCTL.org
37) a15= 32 , d= 7
38) -1, 5, 11, 17, …
Write the recursive formula for the sequence.
39) a1= 4, d= 3
40) a1= 10, d= -8
41) 2, 11, 20, 29, …
42) -17, -6, 5, 16, …
43) 12, 9, 6, 3, …
Write the explicit formula for the sequence.
44) a15= 32 , d= 7
45) -1, 5, 11, 17, …
46) 12, 9, 6, 3, …
47) 45, 31, 17, 3, …
48) 4.3, 5.7, 7.1, 8.5, …
Solve the following questions.
49) What is the common difference when a1=3 and a11=44?
50) What is 20th term if a1=2 and a10= -34?
51) If a1=22 , an=78 ,and d = 8, what is n?
52) Tom works at a car dealership selling cars. If he makes $4000 a month and $250 per car he sells, how much did he
make in July if he sold 16 cars? In August he made $10,000, how many cars did he sell?
Find the missing term(s) in the sequence.
53) 4, _ , 16
54) 3, _ , _ , 27
55) 12, _ , _ , -9
56) 12, _ , _ , _, 48
Geometric Sequences
Class Work
Find the common ratio in sequence, and then write the next 3 terms in the sequence.
57) 5, 10, 20, 40, …
58) 4, -12, 36, -108, …
59) 16, -8, 4, -2, …
60) 6, 9, 13.5, 20.25, …
Use the given information to write the first 4 terms of the sequence and the 10 th term.
61) a1=6 and r=3
62) a1=12 and r=.5
63) a1=8 and r= 4
64) a3=20 and r=2
65) A cell reproduces by splitting in half every half hour, how many cells will there be in 6 hours if a the start there were
10?
Write the recursive formula for the sequence.
66) a1=6 and r=3
67) a1=12 and r=.5
68) a1=8 and r= 4
69) 5, 10, 20, 40, …
70) 4, -12, 36, -108, …
Write the explicit formula for the sequence.
71) 16, -8, 4, -2, …
72) 6, 9, 13.5, 20.25, …
73) a1=8 and r= 4
74) a3=20 and r=2
Alg II- Sequences & Series
~2~
NJCTL.org
Find the missing term(s) in the sequence.
75) 4, _ , 144
76) 5, _ , _, 135
77) 6, _ , _, -.75
78) 7, _ , _ , _ , 4375
Homework
Find the common ratio in sequence, and then write the next 3 terms in the sequence.
79) 2, 8, 32, 128, …
80) 6, -12, 24, -48, …
81) 20, -10, 5, -2.5, …
82) -4, 5, -6.25, 7.8125 , …
Use the given information to write the first 4 terms of the sequence and the 10 th term.
83) a1=8 and r=10
84) a1=20 and r=.5
85) a1=16 and r= -2
86) a3=36 and r=3
87) A cell reproduces by splitting in half every 15 minutes, how many cells will there be in 5 hours if a the start there
were 20?
Write the recursive formula for the sequence.
88) a1=8 and r=10
89) a1=20 and r=.5
90) 2, 8, 32, 128, …
91) 6, -12, 24, -48, …
Write the explicit formula for the sequence.
92) a1=16 and r= -2
93) a3=36 and r=3
94) 20, -10, 5, -2.5, …
95) -4, 5, -6.25, 7.8125 , …
Find the missing term(s) in the sequence.
96) 4, _ , 36
97) 5, _ , _, 1080
98) 6, _ , _, 3072
99) 7, _ , _ , _ , 1792
Geometric Series
Class Work
Find the indicated Sn.
100) a1=7, r=3, and n= 8, find S8
101) a1=6, r= -2, and n= 12, find S12
102) a1=8, r=.5, and n= 6 find S6
103) a1=4 and a8 = 312500, find S8
104) a1=4 and a8 = 312500, find S6
Evaluate the following
105) ∑4𝑛=1 2 ∙ 3𝑛−1
106) ∑5𝑛=1 3 ∙ 2𝑛−1
107) ∑6𝑛=1 −2 ∙ (−2)𝑛−1
108) ∑5𝑛=1 6 ∙ (−.5)𝑛−1
Homework
Find the indicated Sn.
109) a1=6, r=5, find S9
Alg II- Sequences & Series
~3~
NJCTL.org
110) a1=10, r= -4, find S7
111) a1=48, r=.5, find S8
112) a1=12 and a8 = 26244, find S8
113) a1=12 and a8 = 26244, find S6
Evaluate the following
114) ∑4𝑛=1 6 ∙ 3𝑛−1
115) ∑6𝑛=1 8 ∙ (4)𝑛−1
116) ∑7𝑛=1 −10 ∙ (1.5)𝑛−1
117) ∑8𝑛=1 12 ∙ (−2)𝑛−1
Special Sequences
Class Work
Write the first four terms of the sequence.
118) a1=11, 𝑎𝑛 = −2𝑎𝑛−1 + 4
119) a1=-3, a2=2, 𝑎𝑛 = 4𝑎𝑛−1 − 3𝑎𝑛−2
Identify the sequence as arithmetic, geometric, or neither. Write the first 5 terms.
120) a1=3; an= 4an-1 + 2
121) a1=6; an= -3an-1
122) a1=-4; an= an-1 + 8
123) a1=-7; an= 2an-1 -1
124) a1=8; an= an-1 + 2
125) a1=1; an= 6an-1
126) a1=-9;a2=6; an= 4an-1 + 2an-2
127) a1=10; a2= 8; an= -1an-1 + 2an-2
Home Work
Write the first four terms of the sequence.
128) a1=-7; an= 2an-1 -1
129) a1=10; a2= 8; an= -1an-1 + 2an-2
Identify the sequence as arithmetic, geometric, or neither. Write the first 5 terms.
130) a1=2; an= an-1 + 8
131) a1=8; an= -2an-1
132) a1=-14; an= an-1 – 5
133) a1=-1; an= 6an-1
134) a1=12; an= 3an-1
135) a1=5; an=2 + 5an-1
136) a1=-4;a2=3; an= 2an-1 + 3an-2
Writing Sequences as Functions
Class Work
Write the sequence as a function and state the domain.
137) 20, -10, 5, -2.5, …
138) -4, 5, -6.25, 7.8125 , …
139) 12, 9, 6, 3, …
140) 45, 31, 17, 3, …
Home Work
141) 2, 8, 32, 128, …
142) 6, -12, 24, -48, …
143) 16, -8, 4, -2, …
144) 6, 9, 13.5, 20.25, …
Alg II- Sequences & Series
~4~
NJCTL.org
Review
Multiple Choice
1) The sequence 2, 6, 10, 14,… has
a. a common ratio of 3
b. a common difference of 3
c. a common ratio of 4
d. a common difference of 4
2) The sequence 8, -4, 2, -1 ,… has
a. a common ratio of 1/2
b. a common difference of 1/2
c. a common ratio of -1/2
d. a common difference of -1/2
3) The next term in the sequence 7, 13, 19, 25 is
a. 6
b. 31
c. 32
d. 48
4) The next term in the sequence 64, 48, 36, 27 is
a. 20
b. 20.25
c. 20.5
d. 21
5) What is the common difference in this arithmetic sequence: 4, __, __,__, 25
a. 5.25
b. 6.5
c. 7
d. 7.4
6) What is the common ratio in this geometric sequence: 4, __, __, 108
a. ±3
b. 3
c. ±27
d. 27
7) What is the common ratio in this geometric sequence: 32, __, __, __, 2
a. ±.5
b. .5
c. ±2
d. 2
8) a1 = 6 and d = 8 find S6
a. 46
b. 48
c. 108
d. 156
9) a1=8 and r = -4, find S6
a. -10920
b. -8192
c. -6552
d. -12
8
∑
10) n=4 3n =
a. 24
b. 36
c. 72
d. 90
Alg II- Sequences & Series
~5~
NJCTL.org
11) ∑9n=1 2n − 5=
a. 45
b. 40
c. 28
d. 13
1 𝑛−1
12) ∑∞
𝑛=1 4 (− )
2
a. 0
b. 2.5
c. 2 2/3
d. not possible
13) Find the first 5 terms of a1 = 3 and an= 2an-1
a. 3, 6, 9, 12, 15
b. 3, 5, 7, 9, 11
c. 3, 6, 12, 24, 36
d. 3, 6, 12, 24, 48
14) Find the first 5 terms of a1 = 2 and a2=8 and an= an-1 + an-2
a. 2, 8, 10, 18, 28
b. 2, 8, 10, 12, 20
c. 2, 8, 16, 32, 64
d. 2, 8, 16, 24, 32
15) The sequence 1, 1, 2, 3, 5, 8, … is
a. arithmetic
b. geometric
c. Fibonacci
d. None of the above
Extended Response
1) Your rich uncle wants to hire you to work for his company for 30 days. He offers to pay you (A) $1000 per day or (B)
1 cent the first day and double the pay each day their after.
a. How much money do you make in 30 days of pay A?
b. How much do you make on day 30 of pay B?
c. How much money do you make in 30 days of pay B?
2) Given the sequence 2, ___, ___, -10
a. What are the missing terms if the sequence were geometric?
b. What are the missing terms if the sequence were arithmetic?
c. What is the sum of each sequence from part a and b. Use Sigma notation.
3) Given the arithmetic sequence of 4, 10, 16, 22, …
a. What is a20?
b. What is an?
c. What is S20?
d. What is Sn?
4) Given the geometric sequence of 20, -10, 5, -2.5, …
a. What is a10?
b. What is an?
c. What is S20?
d. What is Sn?
e. What is 𝑆∞ ?
Alg II- Sequences & Series
~6~
NJCTL.org
ANSWERS- SEQUENCES AND SERIES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
D=4; 19, 23, 27
2=7; 29, 36, 43
D=-3; -7, -10, -13
D=-12; 20, 8, -4
D=1.3; 6.5, 7.8, 9.1
4, 12, 20, 28, 36; 156
10, 4, -2, -8, -14; -104
3, 9, 15, 21, 27; 117
-38, -33, -28, -23 -18; 57
4, 7, 10, 13, 16; 61
an=an-1+4, a1=3
an=an-1-3, a1=5
an=an-1+1.3, a1=1.3
an=an-1+6, a1=9
an=an-1+5, a1=-38
an=8n-4
an=10n-16
an=6n-3
an=7n-6
an=-12n+80
D=3.2
-83
12
7000; 10 cars
6
9, 14
1, -3
32, 44, 56
d=9; 38, 47, 56
D=11; 27, 38, 49
D=-3; 0, -3, -6
D=-14; -11, -25, -39
D=1.4; 9.9, 11.3, 12.7
4, 7, 10, 13, 16; 61
10, 2, -6, -14, -22; -142
-5, 9, 23, 37, 51; 261
-66, -59, -52, -45, -38; 67
-1, 5, 11, 18, 25; 113
an=an-1+3, a1=4
an=an-1-8, a1=10
an=an-1+9, a1=2
an=an-1+11, a1=-17
an=an-1-3, a1=12
an=7n-73
an=6n-7
an=-3n+15
an=14n-31
an=1.4n+2.9
4.1
-4.2
Alg II- Sequences & Series
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
8
8000 dollars; 24 cars
10
11, 19
5, -2
21, 30, 39
R=2; 80, 160, 320
R=-3; 324, -972, 2916
R=-1/2; 1, -1/2, ¼
R=1.5; 30.375; 45.5625,
68.34375
6, 18, 54, 162; 118098
12, 6, 3, 1.5; .0234375
8, 32, 128, 512; 2097152
5, 10, 20, 40; 2560
40950
an=3an-1, a1=6
an=0.5an-1, a1=12
an=4an-1, a1=8
an=2an-1, a1=5
an=-3an-1, a1=4
an=16(-1/2)n-1
an=6(1.5)n-1
an=8(4)n-1
an=5(2)n-1
+/- 24
15, 45
-3, 1.5
+/-35, 175, +/- 875
R=4; 512, 2048, 8192
R=-2; 96, -192, 384
R=-.5; 1.25, -.625, .3125
R=-1.25; -9.765625,
12.20703125
8, 80, 800, 8000;
8,000,000,000
20, 10, 5, 2.5; .0390625
16, -32, 64, -128; -8192
4, 12, 36, 108; 78732
209 715 20
an=10an-1, a1=8
an=.5an-1, a1=20
an=4an-1, a1=2
an = -2an-1, a1=6
an=16(-2)n-1
an=4(3)n-1
an=20(-0.5)n-1
an = -4(-1.25)n-1
+/- 12
30, 180
~7~
98. 48, 384
99. +/- 28, 112, +/- 448
100. 22960
101. -8190
102. 15.75
103. 390,624
104. 15,624
105. 80
106. 93
107. 42
108. 4.125
109. 2,929,686
110. 32,770
111. 95.625
112. 39360
113. 4368
114. 240
115. 10920
116. -321.71875
117. -1020
118. 11, -18, 40, -76
119. -3, 2, 17, 62
120. N; 3, 14, 58, 234, 938
121. G; 6, -18, 54, -162, 486
122. A; -4, 4, 12, 20, 28
123. N; -7, -15, -31, -63, -127
124. A; 8, 10, 12, 14, 16
125. G; 1, 6, 36, 216, 1296
126. N; -9, 6, 6, 36, 156
127. N; 10, 8, 12, 4, 20
128. -7, -15, -31, -63
129. 10, 8 , 12, 4
130. A; 2, 10, 18, 26, 34
131. G; 8, -16, 32, -64, 128
132. A; -14, -19, -24, -27, -34
133. G; -1, -6, -36, -216, -1296
134. G; 12, 36, 108, 324, 972
135. N; 5, 27, 137, 687, 3437
136. N; -4, 3, -6, -3, =24
137. f(x)=20(-1/2)x-1, all real #’s
138. f(x)=-4(-1/4)x-1, all real #’s
139. f(x)=-3x+15, all real #’s
140. f(x)=-4x+59, all real #’s
141. f(x)=2(4)x-1, all real #’s
142. f(x)=6(-2)x-1, all real #’s
143. f(x)=16(-1/2)x-1, all real #’s
144. f(x)=6(1.5)x-1, all real #’s
NJCTL.org
Multiple Choice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
D
C
B
B
A
B
A
D
C
D
A
C
D
A
C
Extended Response
1.
a. 30000
b. 5,368,709
c. 10,737,418.23
2.
a. -3.42, 5.848
b. -2, -6
c.
3
𝑛
∑4𝑛=1 2 √−5 = −5.572
4
∑ 6 − 4𝑛 = −16
𝑛=1
3.
a.
b.
c.
d.
118
an=6n-2
1220
𝑆𝑛 = 17𝑛 + 3𝑛2
4.
a. -0.039
1
b. 𝑎𝑛 = 20(− )𝑛
2
c.
𝑆20 = 13.33
𝑛
d. 𝑆𝑛 = 13.33(1 − (−12) )
e. 𝑆∞ =
40
3
Alg II- Sequences & Series
~8~
NJCTL.org