Contemporary Mathematics

# 3.11Geometric Sequences

Contemporary Mathematics3.11 Geometric Sequences

Figure 3.49 Savings grows in a geometric sequence. (credit: modification of “A big part of financial freedom is having your heart and mind free from worry about the what-ifs of life. – Suze Orman” by Morgan/Flickr, CC BY 2.0)

### Learning Objectives

After completing this section, you should be able to:

1. Identify geometric sequences.
2. Find a given term in a geometric sequence.
3. Find the $nn$th term of a geometric sequence.
4. Find the sum of a finite geometric sequence.
5. Use geometric sequences to solve real-world applications.

1.

### Section 3.11 Exercises

For the following exercises, determine if the sequence is a geometric sequence.
1 .
$\left\{ {3,7,11,15,25,100,...} \right\}$
2 .
$\left\{ {2,4,8,16,32,...} \right\}$
3 .
$\left\{ {9,0.9,0.09,0.009,0.00009,...} \right\}$
4 .
$\left\{ {\text{262,144},\text{65,536},\text{16,384},\text{4,096},\text{1,024},...} \right\}$
5 .
$\left\{ {14,19,24,29,34,50,60} \right\}$
6 .
$\left\{ {3.9,2.3,0.7, - 0.9, - 2.5, - 4.1, - 5.7,..} \right\}$
7 .
$\left\{ {4, - 8,16, - 32,64, - 128,256,...} \right\}$
8 .
$\left\{ {8, - 4,2, - 1,0.5, - 0.25,0.125, - 0.0625,...} \right\}$
For the following exercises, the sequences given are geometric sequences. Determine the common ratio for each. Verify that each term is the previous term times the common ratio.
9 .
$\left\{ {3,6,12,24,48,96,...} \right\}$
10 .
$\left\{ {8,24,72,216,648,1944,...} \right\}$
11 .
$\left\{ {15,3,0.6,0.12,0.024,0.0048,0.00096,...} \right\}$
12 .
$\left\{ {52,26,13,6.5,3.25,1.625,0.8125,0.40265...} \right\}$
13 .
$\left\{ {18, - 18,18, - 18,18 - 18,...} \right\}$
14 .
$\left\{ {48, - 12,3 - 0.75,0.1875, - 0.046875,...} \right\}$
For the following exercises, the first term and the common ratio of a geometric sequence is given. Using that information, determine the indicated term of the sequence.
15 .
${a_1} = 5$, $r = 3$, find ${a_6}$.
16 .
${b_1} = 7$, $r = 9$, find ${b_5}$.
17 .
${c_1} = 11$, $r = 4$, find ${c_{12}}$.
18 .
${a_1} = 2$, $r = 7$, find ${a_9}$.
19 .
${t_1} = 100$, $r = \frac{1}{5}$, find ${t_{10}}$.
20 .
${b_1} = 56$, $r = 0.25$, find ${b_{15}}$.
21 .
${b_1} = 13$, $r = - 2$, find ${b_{10}}$.
22 .
${a_1} = 11$, $r = - 3$, find ${a_{12}}$.
23 .
${a_1} = 12$, $r = - \frac{1}{3}$, find ${a_8}$.
24 .
${a_1} = 100$, $r = - 10$, find ${a_{15}}$.
For the following exercises, the first term and the common ratio is given for a geometric sequence. Use that information to find the sum of the first $n$ terms of the sequence, ${s_n}$.
25 .
${a_1} = 3$, $r = 4$, calculate ${s_5}$.
26 .
${a_1} = 5$, $r = 3$, calculate ${s_9}$.
27 .
${a_1} = 4$, $r = 5$, calculate ${s_8}$.
28 .
${a_1} = 48$, $r = 2$, calculate ${s_{11}}$.
29 .
${a_1} = 450$, $r = 0.5$, calculate ${s_{12}}$.
30 .
${a_1} = 300$, $r = 0.25$, calculate ${s_{10}}$.
31 .
${a_1} = 3$, $r = - 2$, calculate ${s_{11}}$.
32 .
${a_1} = 5$, $r = - 4$, calculate ${s_8}$.
For the following exercises, apply your understanding of geometric sequences to real-world applications.
33 .
Lactobacilius acidophilus (L. acidophilus) is a bacterium that grows in milk. In optimal conditions, its population doubles every 26 minutes. If a culture starts with 20 L. acidophilus bacteria, how many bacteria will there be after 390 minutes? Hint: This means the 26-minute time period has occurred 15 times.
34 .
Bacillus megaterium (B. megaterium) is a bacterium that grows in sucrose salts. In optimal conditions, its population doubles every 25 minutes. If a culture starts with 30 B. megaterium bacteria, how many bacteria will there be after 1,000 minutes? Hint: This means the 25-minute time period has occurred 40 times.
35 .
Alex and Jill deposit $4,000 in an account bearing 5% interest compounded yearly. If they do not deposit any more money in that account, how much will it be worth in 30 years? 36 . Kerry and Megan deposit$6,000 dollars in and account bearing 4% compounded yearly. If they do not deposit any more money in that account, how much will be in the account after 40 years?
37 .
You decide to color a square that measures 1 m on each side in a very particular manner. You first cut the square in half vertically. You color one side of the square with purple. On the side of the square that was not colored, you draw a line dividing that region horizontally exactly in half. You color the lower half blue. Now, you cut the remaining quarter of the square precisely in half with a vertical line. You color the left side red. You repeat this process 12 times. After you color that 12th piece, what is the total area you have colored?
38 .
Consider the geometric sequence with first term 0.9 and common ratio of 0.1. What is the sum of the first 5 terms?
39 .
Repeat Exercise 38, for the sum of the first 10 terms.
For the following questions, recall that the formula for interest compounded yearly is $A = P{\left( {1 + r} \right)^t}$, where $A$ is the amount in the account after $t$ years, $P$ is the initial amount deposited, and $r$ is the interest rate per year. However, if the account is compounded monthly, the formula changes to $A = P{\left( {1 + \frac{r}{{12}}} \right)^{12t}}$.
40 .
Returning to Kerry and Megan (Exercise 36), what would their account be worth if their account was compounded monthly?
41 .
Returning to Alex and Jill (Exercise 35), what would their account be worth if their account was compounded monthly?
42 .
Imagine your family tree. You have two parents. Your parents have two parents: your grandparents. And so on. How many great-great-great-great-grandparents do you have? Hint: This would be six generations back.
43 .
Imagine your family tree. You have two parents. Your parents have two parents: your grandparents. And so on. How many great (20 times) grandparents do you have? Hint: This would be 22 generations back.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.