Alg 2 Sequences and Series
Review #1
Name
Per
Arithmetic
Geometric
Recursive:
𝑎1 = 𝑘 𝑎𝑛 = 𝑎𝑛−1 + 𝑑
Recursive:
𝑎1 = 𝑘 𝑎𝑛 = 𝑟 ∙ 𝑎𝑛−1
Explicit:
𝑎𝑛 = 𝑎1 + 𝑑(𝑛 − 1)
Explicit:
𝑎𝑛 = 𝑎1 ∙ (𝑟)𝑛−1
𝑛
𝑆𝑛 = 2 (𝑎1 + 𝑎𝑛 )
𝑆𝑛 =
𝑎1 −𝑟∙𝑎𝑛
1−𝑟
𝑆𝑛 =
𝑎1 (1−𝑟 𝑛 )
1−𝑟
𝑎
1
𝑆∞ = 1−𝑟
For each sequence below, describe the pattern, find the next term, and determine if the sequence represents
arithmetic, geometric, or other sequence.
1.
2, 8, 32, 128, …
2.
2, 5, 10, 17, 26, …
3.
12, 10.2, 8.4, 6.6, …
Description:
Description:
Description:
Next Term:
Next Term:
Next Term:
Arithmetic/Geometric/Other
Arithmetic/Geometric/Other
Arithmetic/Geometric/Other
For each sequence below, determine if it is arithmetic or geometric. Then write a recursive formula and an
explicit formula for each.
1.
6, 11, 16, 21, 26, …
5.
7, -14, 28, -56, …
6.
15.5, 12.5, 9.5, 6.5, …
Arithmetic/Geometric
Arithmetic/Geometric
Arithmetic/Geometric
Recursive:
Recursive:
Recursive:
Explicit:
Explicit:
Explicit:
For questions 7 and 8, identify the common difference or common ratio
7.
12.6, 18.3, 24, 29.7, …
8.
Common difference:
12, 18, 27, 40.5, …
Common ratio:
For questions 9 and 10, do the following.
a) Fill in the blanks to make the sequence arithmetic.
b) Fill in the blanks to make the sequence geometric.
9. a) 4, 36, _____ ,…
b) 4, 36, _____ ,…
10. a) 2,
b)
2,
, 50, …
, 50, …
11. A arithmetic sequence has the following two terms: 𝑎3 = 7 and 𝑎10 = −21 . Write the explicit formula
defining the value of the 𝑛𝑡ℎ term 𝑎𝑛 .
12. A geometric sequence has the following two terms: 𝑎2 = −6 and 𝑎7 = 192. Write the explicit formula
defining the value of the 𝑛𝑡ℎ term 𝑎𝑛 .
First identify if the series is arithmetic, geometric, or neither. Write the explicit formula. Write the series in sigma
notion. Finally, find the sum. Show your formulas!
13. 2 + 5 + 8 + 11 + 14 … 98
16. −7 − 10.2 − 13.4 − 16.6 − ⋯ − 407
Arithmetic/Geometric/Neither
Arithmetic/Geometric/Neither
Explicit:
Explicit:
Sigma Notation:
Sigma Notation:
Sum:
Sum:
14. 6 − 12 + 24 − 48 + 96 − 192 + 384 − 768
17. 2 + 5 + 10 + 17 + 26 + ⋯ + 101
Arithmetic/Geometric/Neither
Arithmetic/Geometric/Neither
Explicit:
Explicit:
Sigma Notation:
Sigma Notation:
Sum:
Sum:
15. 16 + 24 + 36 + 54 + ⋯ +
Arithmetic/Geometric/Neither
Explicit:
Sigma Notation:
Sum:
2187
8
18. 64 + 16 + 4 + 1 + ⋯ and has 12 terms
Arithmetic/Geometric/Neither
Explicit:
Sigma Notation:
Sum:
Write out the first five terms of each sequence or series.
19. 𝑎1 = −15 ; 𝑎𝑛 = 𝑎𝑛−1 + 6
12
20. 𝑎1 = 5 ; 𝑎𝑛 = −3 ∙ 𝑎𝑛−1
21. 𝑎1 = 10 ; 𝑎𝑛 = 2 ∙ 𝑎𝑛−1 − 6
12
150
22. ∑ 8 + 5(𝑛 − 1)
23. ∑ 2(−3)𝑛−1
𝑛=1
24.
𝑛=4
𝑛=1
Find the sum of each series.
25. a)
∑ 5𝑛 − 12
b)
12
c)
12
150
𝑛−1
∑ 8 + 5(𝑛 − 1)
∑ 2(−3)
∑ 5𝑛 − 12
𝑛=1
𝑛=1
𝑛=4
Find the sum of each infinite geometric series, if possible. (Bonus)
26. a)
∞
b)
𝑛−1
∞
c)
𝑛−1
∑ 5(−0.8)
∑ 4(−3)
𝑛=1
𝑛=1
∞
3 𝑛−1
∑ −4 ( )
4
𝑛=3
27. You are in an auditorium and sitting in row 12. You notice the row you are sitting in has 65 seats and the row
behind you has 68 seats while the row in front of you has 62 seats. You also notice there are 35 rows in the
auditorium
a. How many seats are in the first row?
b. How many seats in the last row?
c. How many total seats in the auditorium?
28. A beginning teacher earns $42,500 if the teacher has a master’s degree. The district is planning to fund the
cost of living increases of 3% each year, which will be a rate of 1.03.
a. What will be the teacher’s salary when the teacher retires in 28 years?
b. What will be the total amount of money the school district will need to pay the teacher during the
teacher’s career?
29. A computer programmer earns $65,000 in their first year from college. The company they work for typically
raises their salary 5% each year.
a. How much will the computer programmer earn in their 20th year of employment?
b. How much will they earn over their entire time with the company?