Geometric Sequences

Show your work for each problem.

1.
Which of these is a geometric sequence?
1. a)
2. b)
3. c)
4. d)
Answer:
is geometric because each term is multiplied by . The others are: arithmetic (+2, +5), and perfect squares.
2.
What is the common ratio of the sequence ?
Answer: 3
. Each term is 3 times the previous term.
3.
In the geometric sequence , what is the next term?
Answer: 162
The ratio is . The next term is .
4.
What is the common ratio of ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This is a decreasing geometric sequence (exponential decay).
5.
Find the 6th term of the geometric sequence: ,
Answer: 96
1. What formula do we use to find the nth term? a_n = a_1 * r^(n-1)
2. What is the exponent for the 6th term? (n-1 where n=6) 5
3. Calculate 32
4. Calculate 96
6.
Find the 5th term of the sequence
Answer: 80
, .
7.
A bacteria colony starts with 500 cells and doubles every hour. How many bacteria are there after 4 hours?
1. a)
2. b)
3. c)
4. d)
Answer:
bacteria.
8.
Find the 7th term of the sequence
Answer: 729
1. What is the first term ? 1
2. What is the common ratio ? 3
3. What is ? 729
4. Calculate 729
9.
You invest 1000 euros at 10% annual interest. How much will you have after 3 years? (Round to the nearest euro)
Answer: 1331
euros.
10.
A ball is dropped from 128 cm. Each bounce reaches half the previous height. What height does it reach after the 4th bounce?
Answer: 8
1. What is the initial height (first term)? 128
2. What is the common ratio? 0.5
3. After 4th bounce is which term? (bounce 0 = term 1) 5
4. Calculate 8
11.
What is the common ratio of the sequence ?
1. a)
2. b)
3. c)
4. d)
Answer:
. The negative ratio causes the signs to alternate.
12.
In a geometric sequence, and . Find the common ratio.
Answer: 3
1. Write in terms of : 1
2. Write in terms of : 4
3. Divide: 27
4. This equals . What is ? 3
13.
Find the first term of a geometric sequence where and .
Answer: 8
. So .
14.
A car worth 20000 euros depreciates by 15% each year. What is it worth after 4 years? (Round to the nearest euro)
Answer: 10440
1. If it loses 15%, what fraction does it keep? 0.85
2. What is the common ratio? 0.85
3. After 4 years is which term? (year 0 = term 1) 5
4. Calculate (round to nearest euro) 10440
15.
In a geometric sequence, and . Find the first term .
Answer: 2
, so . Then , so .
16.
Find the 10th term of the sequence
Answer: 39366
1. What is the first term? 2
2. What is the common ratio? 3
3. What is the exponent for the 10th term? 9
4. Calculate 39366