Arithmetic Sequences

Show your work for each problem.

1.
Which sequence is arithmetic?
1. a)
2. b)
3. c)
4. d)
Answer:
has a constant difference of between each term. The first is geometric, the third is Fibonacci, and the fourth is perfect squares.
2.
Find the common difference of:
Answer: -3
. Each term decreases by , so .
3.
In the sequence , what is the common difference ?
Answer: 4
. We can verify: and .
4.
What is in the sequence ?
1. a)
2. b)
3. c)
4. d)
Answer:
because is the first term of the sequence. The subscript indicates the position, and 8 is in position 1.
5.
Find the 10th term of the sequence:
Answer: 47
1. What is the first term ? 2
2. What is the common difference ? 5
3. Using , what is when ? 9
4. Calculate 47
6.
Find for the sequence:
Answer: 118
, . So .
7.
If and , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
8.
Find the 15th term of:
Answer: 2
1. What is ? 100
2. Calculate the common difference -7
3. For , what is ? 14
4. Calculate 2
9.
What is the 50th term if and ?
Answer: 101
10.
In the sequence , which term equals ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Solving : , so , thus . But wait - let's verify: . Actually checking: for . So gives , , .
11.
If and , find and .
Answer: a_1 = 5, d = 4
1. Write the equation for : 17
2. Write the equation for : 37
3. Subtract the first equation from the second to get 20
4. What is ? 4
5. Substitute back: , so 5
12.
A theater has 15 seats in the first row and each row has 4 more seats than the previous. How many seats are in row 20?
Answer: 91
seats.
13.
You start saving 200 euros and add 75 euros each month. After how many months will you have 1250 euros?
Answer: 15
1. What is (starting amount)? 200
2. What is (monthly addition)? 75
3. Set up: . What is ? 1050
4. What is ? 14
5. What is ? 15
14.
If and , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
From positions 3 to 7, there are 4 steps. , so . Then . Finally, .
15.
Find the value of if form an arithmetic sequence.
Answer: 7
For arithmetic sequences, the middle term equals the average of adjacent terms: .
16.
What type of sequence is ?
1. a)Arithmetic with
2. b)Not arithmetic
3. c)Arithmetic with
4. d)Arithmetic with
Answer: Arithmetic with
Since , the common difference is . This is still arithmetic because the difference is constant!
17.
Write the explicit formula for the sequence:
Answer: a_n = -3 + (n-1) * 4
1. What is ? -3
2. Calculate 4
3. Write the formula: . What is ? -3
4. Complete: 4