Introduction to Sequences

Show your work for each problem.

1.
What is the next term in the sequence ?
1. a)14
2. b)12
3. c)13
4. d)15
Answer: 14
The pattern adds 3 each time: , , . So the next term is .
2.
Which type of sequence is ?
1. a)Neither
2. b)Fibonacci
3. c)Arithmetic
4. d)Geometric
Answer: Geometric
Each term is multiplied by 2: , , . This is a geometric sequence with ratio 2.
3.
In the sequence , what is the common difference?
Answer: 3
The common difference is , , . Each term increases by 3.
4.
In the sequence , what is the common ratio?
Answer: 3
The common ratio is , , . Each term is multiplied by 3.
5.
Find the 5th term of the sequence
Answer: 13
1. What is the first term ? 1
2. What is the common difference ? 3
3. Using , calculate 13
6.
Find the next three terms:
Answer: 22, 26, 30
1. What is the common difference? 4
2. What is the 5th term? () 22
3. What is the 6th term? 26
4. What is the 7th term? 30
7.
What is the formula for the th term of ?
1. a)
2. b)
3. c)
4. d)
Answer:
, . So .
8.
What is the 6th term of the geometric sequence ?
1. a)648
2. b)324
3. c)162
4. d)486
Answer: 486
.
9.
Find the 10th term of the sequence
Answer: 48
.
10.
Find the 5th term of the geometric sequence
Answer: 324
.
11.
Write the formula for the th term of
Answer: 5n + 1
1. What is the first term ? 6
2. What is the common difference ? 5
3. Write and simplify 5n + 1
12.
Find the 8th term of the sequence
Answer: 128
1. What is the first term ? 1
2. What is the common ratio ? 2
3. What is ? 128
4. Calculate 128
13.
If and in an arithmetic sequence, what is ?
1. a)4
2. b)3
3. c)5
4. d)2
Answer: 2
, so , thus , and .
14.
Find the 20th term of
Answer: 71
.
15.
In a geometric sequence, and . Find the common ratio.
Answer: 3
, so , thus .
16.
Find the first term of an arithmetic sequence where and .
Answer: 4
1. Write the formula: a_1 + 3d
2. Substitute: a_1 + 15
3. Solve for : 4
17.
A bacteria population doubles every hour, starting with 500. How many after 10 hours?
Answer: 256000
1. What type of sequence is this? geometric
2. What is (initial population)? 500
3. What is the common ratio ? 2
4. Calculate 512
5. Calculate 256000
18.
Which term of the sequence equals 99?
Answer: 24
1. What is ? 7
2. What is the common difference ? 4
3. Set up the equation: 99 = 7 + 4n - 4
4. Simplify: . Solve for . 24