College Algebra and Trigonometry

Margaret L. Lial, John Hornsby, David I. Schneider

Chapter 11 Further Topics in Algebra - all with Video Answers

Educators

Section 1

Sequences and Series

00:43

Problem 1

$\mathrm{A}(\mathrm{n})$ _________ is a function that computes an ordered list.

Yujie Wang

College of San Mateo

00:42

Problem 2

A(n) _________ sequence is a function that has the set of natural numbers of the form $\{1,2,3, \ldots, n\}$ as its domain.

Yujie Wang

College of San Mateo

00:48

Problem 3

Some sequences are defined by a(n) _________ definition, one in which each term after the first term or the first few terms is defined as an expression involving the previous term or terms.

Yujie Wang

College of San Mateo

00:40

Problem 4

The sum of the terms of a sequence is a(n) _____________. It is written using the Greek capital letter symbol _______ to indicate a sum.

Yujie Wang

College of San Mateo

01:47

Problem 5

Complete a table of values for the sequence $a_{n}=5 n+2$ using $n=1,2,3,4,5 .$

Yujie Wang

College of San Mateo

00:47

Problem 6

Graph the sequence $a_{n}=5 n+2$ using the values from Exercise 5

James Kiss

Numerade Educator

02:00

Problem 7

$$
\text { Evaluate } \sum_{i=1}^{5}(5 i+2)
$$

Yujie Wang

College of San Mateo

01:26

Find the first five terms of the sequence defined by the following recursive definition. How is the sequence related to the sequence in Exercise $5 ?$
$$\begin{array}{l}a_{1}=7 \\a_{n}=a_{n-1}+5, \quad \text { if } n>1\end{array}$$

Yujie Wang

College of San Mateo

03:12

Problem 9

Find the first five terms of the sequence $a_{n}=3(-3)^{n-1}$.

Chris Wojturski

Numerade Educator

02:52

Problem 10

Evaluate $\sum_{i=1}^{5} 3(-3)^{i-1}$.

Chris Wojturski

Numerade Educator

00:57

Problem 11

Write the first five terms of each sequence.
$a_{n}=4 n+10$

Rakesh Kumar Sharma

Numerade Educator

00:59

Problem 12

Write the first five terms of each sequence.
$a_{n}=6 n-3$

Rakesh Kumar Sharma

Numerade Educator

01:15

Problem 13

Write the first five terms of each sequence.
$a_{n}=\frac{n+5}{n+4}$

Rakesh Kumar Sharma

Numerade Educator

01:17

Problem 14

Write the first five terms of each sequence.
$a_{n}=\frac{n-7}{n-6}$

Rakesh Kumar Sharma

Numerade Educator

01:22

Problem 15

Write the first five terms of each sequence.
$a_{n}=\left(\frac{1}{3}\right)^{n}(n-1)$

Rakesh Kumar Sharma

Numerade Educator

00:45

Problem 16

Write the first five terms of each sequence.
$a_{n}=\left(\frac{1}{2}\right)^{n}(n)$

Maninder Singh

Numerade Educator

01:06

Problem 17

Write the first five terms of each sequence.
$a_{n}=(-1)^{n}(2 n)$

Rakesh Kumar Sharma

Numerade Educator

01:26

Problem 18

Write the first five terms of each sequence.
$a_{n}=(-1)^{n-1}(n+1)$

Rakesh Kumar Sharma

Numerade Educator

01:59

Problem 19

Write the first five terms of each sequence.
$a_{n}=\frac{4 n-1}{n^{2}+2}$

Rakesh Kumar Sharma

Numerade Educator

01:24

Problem 20

Write the first five terms of each sequence.
$a_{n}=\frac{n^{2}-1}{n^{2}+1}$

Rakesh Kumar Sharma

Numerade Educator

01:55

Problem 21

Write the first five terms of each sequence.
$a_{n}=\frac{n^{3}+8}{n+2}$

Rakesh Kumar Sharma

Numerade Educator

02:17

Problem 22

Write the first five terms of each sequence.
$a_{n}=\frac{n^{3}+27}{n+3}$

Rakesh Kumar Sharma

Numerade Educator

00:40

Problem 23

Decide whether each sequence is finite or infinite.
The sequence of days of the week

Ankit Gupta

Numerade Educator

00:52

Problem 24

Decide whether each sequence is finite or infinite.
The sequence of pages in a book

Rakesh Kumar Sharma

Numerade Educator

00:37

Problem 25

Decide whether each sequence is finite or infinite.
$$
1,2,3,4,5
$$

Rakesh Kumar Sharma

Numerade Educator

00:29

Problem 26

Decide whether each sequence is finite or infinite.
$$
-1,-2,-3,-4,-5
$$

Rakesh Kumar Sharma

Numerade Educator

00:33

Problem 27

Decide whether each sequence is finite or infinite.
$$
1,2,3,4,5, \ldots
$$

Rakesh Kumar Sharma

Numerade Educator

00:39

Problem 28

Decide whether each sequence is finite or infinite.
$$
-1,-2,-3,-4,-5, \ldots
$$

Rakesh Kumar Sharma

Numerade Educator

01:28

Problem 29

Decide whether each sequence is finite or infinite.
$$
\begin{array}{l}
a_{1}=4 \\
a_{n}=4 \cdot a_{n-1}, \text { if } n \geq 2
\end{array}
$$

Rakesh Kumar Sharma

Numerade Educator

01:31

Problem 30

Decide whether each sequence is finite or infinite.
$$
\begin{array}{l}
a_{1}=2 \\
a_{2}=5 \\
a_{n}=a_{n-1}+a_{n-2}, \text { if } n \geq 3
\end{array}
$$

Rakesh Kumar Sharma

Numerade Educator

01:01

Problem 31

Find the first four terms of each sequence. See Example 2.
$$
\begin{array}{l}
a_{1}=-2 \\
a_{n}=a_{n-1}+3, \text { if } n>1
\end{array}
$$

Yujie Wang

College of San Mateo

01:09

Problem 32

Find the first four terms of each sequence. See Example 2.
$$
\begin{array}{l}
a_{1}=-1 \\
a_{n}=a_{n-1}-4, \text { if } n>1
\end{array}
$$

Yujie Wang

College of San Mateo

00:39

Problem 33

Find the first four terms of each sequence. See Example 2.
$$
\begin{aligned}
&a_{1}=1\\
&a_{2}=1\\
&a_{n}=a_{n-1}+a_{n-2}, \text { if } n \geq 3\\
&\text { (This is the Fibonacci sequence.) }
\end{aligned}
$$

Yujie Wang

College of San Mateo

00:59

Problem 34

Find the first four terms of each sequence. See Example 2.
$$
\begin{aligned}
&a_{1}=1\\
&a_{2}=3\\
&a_{n}=a_{n-1}+a_{n-2}, \text { if } n \geq 3\\
&\text { (This is the Lucas sequence.) }
\end{aligned}
$$

Yujie Wang

College of San Mateo

01:01

Problem 35

Find the first four terms of each sequence. See Example 2.
$$
\begin{array}{l}
a_{1}=2 \\
a_{n}=n \cdot a_{n-1}, \text { if } n>1
\end{array}
$$

Yujie Wang

College of San Mateo

01:25

Problem 36

Find the first four terms of each sequence. See Example 2.
$$
\begin{array}{l}
a_{1}=-3 \\
a_{n}=2 n \cdot a_{n-1}, \text { if } n>1
\end{array}
$$

Yujie Wang

College of San Mateo

00:49

Problem 37

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{5}(2 i+1)
$$

Yujie Wang

College of San Mateo

01:18

Problem 38

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{6}(3 i-2)
$$

Yujie Wang

College of San Mateo

00:49

Problem 39

Evaluate each series. See Example 4.
$$
\sum_{j=1}^{4} j^{-1}
$$

Yujie Wang

College of San Mateo

00:56

Problem 40

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{5}(i+1)^{-1}
$$

Yujie Wang

College of San Mateo

00:48

Problem 41

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{4} i^{i}
$$

Yujie Wang

College of San Mateo

00:53

Problem 42

Evaluate each series. See Example 4.
$$
\sum_{k=1}^{4}(k+1)^{k}
$$

Yujie Wang

College of San Mateo

00:49

Problem 43

Evaluate each series. See Example 4.
$$
\sum_{k=1}^{6}(-1)^{k} \cdot k
$$

Yujie Wang

College of San Mateo

02:49

Problem 44

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2}
$$

Yujie Wang

College of San Mateo

00:45

Problem 45

Evaluate each series. See Example 4.
$$
\sum_{i=2}^{5}(6-3 i)
$$

Yujie Wang

College of San Mateo

00:53

Problem 46

Evaluate each series. See Example 4.
$$
\sum_{i=3}^{7}(5 i+2)
$$

Yujie Wang

College of San Mateo

00:47

Problem 47

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{3} 2(3)^{i}
$$

Yujie Wang

College of San Mateo

00:53

Problem 48

Evaluate each series. See Example 4.
$$
\sum_{i=-1}^{2} 5(2)^{i}
$$

Yujie Wang

College of San Mateo

01:44

Problem 49

Evaluate each series. See Example 4.
$$
\sum_{i=-1}^{5}\left(i^{2}-2 i\right)
$$

Yujie Wang

College of San Mateo

01:19

Problem 50

Evaluate each series. See Example 4.
$$
\sum_{i=3}^{6}\left(2 i^{2}+1\right)
$$

Yujie Wang

College of San Mateo

01:32

Problem 51

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{5}\left(3^{i}-4\right)
$$

Yujie Wang

College of San Mateo

01:11

Problem 52

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{4}\left[(-2)^{i}-3\right]
$$

Yujie Wang

College of San Mateo

00:34

Problem 53

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{3}\left(i^{3}-i\right)
$$

Yujie Wang

College of San Mateo

01:07

Problem 54

Evaluate each series. See Example 4.
$$
\sum_{i=1}^{4}\left(i^{4}-i^{3}\right)
$$

Yujie Wang

College of San Mateo

00:36

Problem 55

Use a graphing calculator to evaluate each series. See Example 4.
$$
\sum_{i=1}^{10}\left(4 i^{2}-5\right)
$$

Yujie Wang

College of San Mateo

00:32

Problem 56

Use a graphing calculator to evaluate each series. See Example 4.
$$
\sum_{i=1}^{10}\left(i^{3}-6\right)
$$

Yujie Wang

College of San Mateo

00:39

Problem 57

Use a graphing calculator to evaluate each series. See Example 4.
$$
\sum_{j=3}^{9}\left(3 j-j^{2}\right)
$$

Yujie Wang

College of San Mateo

00:36

Problem 58

Use a graphing calculator to evaluate each series. See Example 4.
$$
\sum_{k=5}^{10}\left(k^{2}-4 k+7\right)
$$

James Kiss

Numerade Educator

02:00

Problem 59

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{5} x_{i}
$$

Anas Venkitta

Numerade Educator

02:00

Problem 60

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{5}-x_{i}
$$

Anas Venkitta

Numerade Educator

03:01

Problem 61

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{5}\left(2 x_{i}+3\right)
$$

Anas Venkitta

Numerade Educator

01:39

Problem 62

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{4}\left(-3 x_{i}-2\right)
$$

Yujie Wang

College of San Mateo

02:32

Problem 63

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{3}\left(3 x_{i}-x_{i}^{2}\right)
$$

Anas Venkitta

Numerade Educator

00:58

Problem 64

Yujie Wang

College of San Mateo

03:18

Problem 65

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=2}^{5} \frac{x_{i}+1}{x_{i}+2}
$$

Anas Venkitta

Numerade Educator

03:14

Problem 66

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{5} \frac{x_{i}}{x_{i}+3}
$$

Anas Venkitta

Numerade Educator

03:41

Problem 67

Write the terms for each series and evaluate the sum, given that $x_{1}=-2, x_{2}=-1, x_{3}=0$, $x_{4}=1,$ and $x_{5}=2 .$ See Examples $5(a)$ and $5(b) .$
$$
\sum_{i=1}^{4} \frac{x_{i}^{3}+1000}{x_{i}+10}
$$

Anas Venkitta

Numerade Educator

02:14

Problem 68

James Kiss

Numerade Educator

02:52

Problem 69

Write the terms of $\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,$ with $x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,$ and $\Delta x=0.5,$ for each function. Evaluate the sum. See Example $5(c) .$
$$
f(x)=4 x-7
$$

Jocelyn Thammavong

Numerade Educator

02:32

Problem 70

Jocelyn Thammavong

Numerade Educator

02:34

Problem 71

Jocelyn Thammavong

Numerade Educator

02:29

Problem 72

Jocelyn Thammavong

Numerade Educator

02:34

Problem 73

Jocelyn Thammavong

Numerade Educator

02:55

Problem 74

Jocelyn Thammavong

Numerade Educator

00:33

Problem 75

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{100} 6
$$

Yujie Wang

College of San Mateo

00:33

Problem 76

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{20} 5
$$

Yujie Wang

College of San Mateo

01:13

Problem 77

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{15} i^{2}
$$

Yujie Wang

College of San Mateo

00:52

Problem 78

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{50} 2 i^{3}
$$

Yujie Wang

College of San Mateo

00:58

Problem 79

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{5}(5 i+3)
$$

James Kiss

Numerade Educator

01:02

Problem 80

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{5}(8 i-1)
$$

James Kiss

Numerade Educator

02:01

Problem 81

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{5}\left(4 i^{2}-2 i+6\right)
$$

Lauren Shelton

Numerade Educator

01:31

Problem 82

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{6}\left(2+i-i^{2}\right)
$$

Lauren Shelton

Numerade Educator

01:34

Problem 83

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)
$$

Lauren Shelton

Numerade Educator

01:20

Problem 84

Use the summation properties and rules to evaluate each series. See Examples 6 and 7.
$$
\sum_{i=1}^{6}\left(i^{2}+2 i^{3}\right)
$$

Lauren Shelton

Numerade Educator

00:34

Problem 85

Use summation notation to write each series.
$$
\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)}
$$

Rakesh Kumar Sharma

Numerade Educator

00:43

Problem 86

Use summation notation to write each series.
$$
\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\cdots+\frac{5}{1+15}
$$

Rakesh Kumar Sharma

Numerade Educator

01:22

Problem 87

Use summation notation to write each series.
$$
1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{128}
$$

Rakesh Kumar Sharma

Numerade Educator

01:34

Problem 88

Use summation notation to write each series.
$$
1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\cdots-\frac{1}{400}
$$

Rakesh Kumar Sharma

Numerade Educator

00:51

Problem 89

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If it converges, determine the number to which it converges.
$$
a_{n}=\frac{n+4}{2 n}
$$

James Kiss

Numerade Educator

02:29

Problem 90

Jocelyn Thammavong

Numerade Educator

00:48

Problem 91

James Kiss

Numerade Educator

00:38

Problem 92

James Kiss

Numerade Educator

01:01

Problem 93

James Kiss

Numerade Educator

01:00

Problem 94

James Kiss

Numerade Educator

03:46

Problem 95

Solve each problem. See Example 3.
Suppose an insect population density, in thousands per acre, during year $n$ can be modeled by the recursively defined sequence
$$\begin{array}{l}a_{1}=8 \\a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \quad \text { for } n>1\end{array}$$
(a) Find the population for $n=1,2,3$.
(b) Graph the sequence for $n=1,2,3, \ldots, 20 .$ Use the window [0,21] by [0,14] Interpret the graph.

Ankit Gupta

Numerade Educator

04:09

Problem 96

Solve each problem. See Example 3.
One of the most famous sequences in mathematics is the Fibonacci sequence,
$$1,1,2,3,5,8,13,21,34,55, \ldots$$
(Also see Exercise 33.) Male honeybees hatch from eggs that have not been fertilized, so a male bee has only one parent, a female. On the other hand, female honeybees hatch from fertilized eggs, so a female has two parents, one male and one female. The number of ancestors in consecutive generations of bees follows the Fibonacci sequence. Draw a tree showing the number of ancestors of a male bee in each generation following the description given above.

Glenn Degamon

Numerade Educator

05:03

Problem 97

Solve each problem. See Example 3.
If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let $N_{1}$ be the initial number of bacteria cells, $N_{2}$ the number after 40 minutes, $N_{3}$ the number after 80 minutes, and $N_{j}$ the number after $40(j-1)$ minutes. (Data from Hoppensteadt, F., and C. Peskin, Mathematics in Medicine and the Life Sciences, SpringerVerlag.)
(a) Write $N_{j+1}$ in terms of $N_{j}$ for $j \geq 1$.
(b) Determine the number of bacteria after $2 \mathrm{hr}$ if $N_{1}=230$.
(c) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 7,$ where $N_{1}=230 .$ Use the window [0,10] by [0,15,000]
(d) Describe the growth of these bacteria when there are unlimited nutrients.

Cullen Miller

Numerade Educator

03:20

Problem 98

Solve each problem. See Example 3.
Refer to Exercise 97. If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst's model, the number of bacteria $N_{j}$ at time $40(j-1)$ in minutes can be determined by the sequence
$$N_{j+1}=\left[\frac{2}{1+\frac{N_{j}}{K}}\right] N_{j}$$
where $K$ is a constant and $j \geq 1$. (Data from Hoppensteadt, F., and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.)
(a) If $N_{1}=230$ and $K=5000,$ make a table of $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Round values in the table to the nearest integer.
(b) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Use the window [0,20] by [0,6000]
(c) Describe the growth of these bacteria when there are limited nutrients.
(d) Make a conjecture about why $K$ is called the saturation constant. Test the conjecture by changing the value of $K$ in the given formula.

Ankit Gupta

Numerade Educator

02:31

Problem 99

Solve each problem. See Example 3.
The series
$$x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots$$
can be used to approximate the value of $\ln (1+x)$ for values of $x$ in $(-1,1] .$ Use the first six terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator.
(a) $\ln 1.02(x=0.02)$
(b) $\ln 0.97(x=-0.03)$

Lauren Shelton

Numerade Educator

01:40

Problem 100

Solve each problem. See Example 3.
Find the sum of the first six terms of the series
$$\frac{\pi^{4}}{90}=\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\frac{1}{5^{4}}+\cdots+\frac{1}{n^{4}}+\cdots$$
Multiply this result by $90,$ and take the fourth root to obtain an approximation of $\pi .$ Compare this answer to the actual decimal approximation of $\pi$.

Lauren Shelton

Numerade Educator

View

Problem 101

Solve each problem. See Example 3.
The series
$$e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\cdots+\frac{a^{n}}{n !}$$
where $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n,$ can be used to approximate the value of $e^{a}$ for any real number $a$. Use the first eight terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator.
(a) $e$
(b) $e^{-1}$

Claire Rochford

Numerade Educator

04:51

Problem 102

Solve each problem. See Example 3.
The recursively defined sequence
$$\begin{array}{l}a_{1}=k \\a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right), \quad \text { if } n>1\end{array}$$
can be used to compute $\sqrt{k}$ for any positive number $k .$ This sequence was known to Sumerian mathematicians 4000 years ago, and it is still used today. Use this sequence to approximate the given square root by finding $a_{6} .$ Compare the result with the actual value. (Data from Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.)
(a) $\sqrt{2}$
(b) $\sqrt{11}$

James Kiss

Numerade Educator