Precalculus

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

Chapter 11

Further Topics in Algebra - all with Video Answers

Educators

Section 1

Sequences and Series

Problem 1

Fill in the blank(s) to correctly complete each sentence.
A(n)___________is a function that computes an ordered list.

Jocelyn Thammavong

Jocelyn Thammavong

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Problem 2

Fill in the blank(s) to correctly complete each sentence.
A(n)_____________ sequence is a function that has the set of natural numbers of the form $\{1,2,3, \ldots, n\}$ as its domain.

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 3

Fill in the blank(s) to correctly complete each sentence.
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.

Jocelyn Thammavong

Jocelyn Thammavong

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Problem 4

Fill in the blank(s) to correctly complete each sentence.
The sum of the terms of a sequence is a(n) _____________ It is written using the Greek capital letter symbol
_________ to indicate a sum.

Jocelyn Thammavong

Jocelyn Thammavong

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Problem 5

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

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 6

Answer each of the following.
Graph the sequence $a_{n}=5 n+2$ using the values from Exercise 5

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 7

Answer each of the following.
$$\text { Evaluate } \sum_{i=1}^{5}(5 i+2)$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 8

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

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 9

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

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 10

Answer each of the following.
$$\text { Evaluate } \sum_{i=1}^{5} 3(-3)^{i-1}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 11

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 12

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 13

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 14

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 15

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 16

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

Maninder Singh

Numerade Educator

Problem 17

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 18

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

Amy Jiang

Numerade Educator

Problem 19

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 20

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 21

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 22

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 23

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

AG

Ankit Gupta

Numerade Educator

Problem 24

Write the first five terms of each sequence.
The sequence of pages in a book

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 25

Write the first five terms of each sequence.
$$1,2,3,4,5$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 26

Write the first five terms of each sequence.
$$-1,-2,-3,-4,-5$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 27

Write the first five terms of each sequence.
$$1,2,3,4,5, \dots$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 28

Write the first five terms of each sequence.
$$-1,-2,-3,-4,-5, \dots$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 29

Write the first five terms of each sequence.
$$\begin{aligned}
&a_{1}=4\\
&a_{n}=4 \cdot a_{n-1}, \text { if } n \geq 2
\end{aligned}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 30

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 31

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

AG

Ankit Gupta

Numerade Educator

Problem 32

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

AG

Ankit Gupta

Numerade Educator

Problem 33

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

AG

Ankit Gupta

Numerade Educator

Problem 34

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 35

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

AG

Ankit Gupta

Numerade Educator

Problem 36

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 37

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

AG

Ankit Gupta

Numerade Educator

Problem 38

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

AG

Ankit Gupta

Numerade Educator

Problem 39

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 40

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

Amy Jiang

Numerade Educator

Problem 41

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

AG

Ankit Gupta

Numerade Educator

Problem 42

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 43

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

AG

Ankit Gupta

Numerade Educator

Problem 44

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

AG

Ankit Gupta

Numerade Educator

Problem 45

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

AG

Ankit Gupta

Numerade Educator

Problem 46

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

AG

Ankit Gupta

Numerade Educator

Problem 47

Evaluate each series.
$$\sum_{i=-2}^{3} 2(3)^{i}$$

Chris Wojturski

Chris Wojturski

Numerade Educator

Problem 48

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 49

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

AG

Ankit Gupta

Numerade Educator

Problem 50

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

AG

Ankit Gupta

Numerade Educator

Problem 51

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

AG

Ankit Gupta

Numerade Educator

Problem 52

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

AG

Ankit Gupta

Numerade Educator

Problem 53

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 54

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 55

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 56

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 57

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 58

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{5} x_{i}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{5}-x_{i}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{5}\left(2 x_{i}+3\right)$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{4}\left(-3 x_{i}-2\right)$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{3}\left(3 x_{i}-x_{i}^{2}\right)$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 64

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 $x_{5}=2 .$
$$\sum_{i=1}^{3}\left(x_{i}^{2}+x_{i}\right)$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=2}^{5} \frac{x_{i}+1}{x_{i}+2}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$
$$\sum_{i=1}^{5} \frac{x_{i}}{x_{i}+3}$$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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 $x_{5}=2 .$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 68

How can factoring make the work in Exercises $21,22,$ and 67 easier?

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

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

Jocelyn Thammavong

Numerade Educator

Problem 70

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 $5(c)$
$$f(x)=6+2 x$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 71

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 $5(c)$
$$f(x)=2 x^{2}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 72

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 $5(c)$
$$f(x)=x^{2}-1$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 73

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 $5(c)$
$$f(x)=\frac{-2}{x+1}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 74

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 $5(c)$
$$f(x)=\frac{5}{2 x-1}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 75

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 76

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 77

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 78

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 79

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 80

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 81

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 82

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 83

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 84

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 85

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 86

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

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

Rakesh Kumar Sharma

Numerade Educator

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

Rakesh Kumar Sharma

Numerade Educator

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.
$$a_{n}=\frac{n+4}{2 n}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 90

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.
$$a_{n}=\frac{1+4 n}{2 n}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 91

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.
$$a_{n}=2 e^{n}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 92

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.
$$a_{n}=n(n+2)$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 93

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.
$$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 94

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.
$$a_{n}=(1+n)^{1 / n}$$

Jocelyn Thammavong

Jocelyn Thammavong

Numerade Educator

Problem 95

Solve each problem.
Suppose an insect population density, in thousands per acre, during year $n$ can be modeled by the recursively defined sequence
$$\begin{aligned}
&a_{1}=8\\
&a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \quad \text { for } n>1
\end{aligned}$$
(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.

AG

Ankit Gupta

Numerade Educator

Problem 96

Solve each problem.
One of the most famous sequences in mathematics is the Fibonacci sequence, $$1,1,2,3,5,8,13,21,34,55, \dots$$
(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

Problem 97

Solve each problem.
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.
(a) Write $N_{j+1}$ in terms of $N_{j}$ for $j \geq 1$
(b) Determine the number of bacteria after 2 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

Problem 98

Solve each problem.
Verhulst's Model for Bacteria Growth 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
$$N_{j+1}=\left[\frac{2}{1+\frac{N}{K}}\right] N_{j}$$
where $K$ is a constant and $j \geq 1 .$ (Source: 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.

AG

Ankit Gupta

Numerade Educator

Problem 99

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) $\ln 0.97(x=-0.03)$

Chris Wojturski

Chris Wojturski

Numerade Educator

Problem 100

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 $\pi$.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 101

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) $e^{-1}$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 102

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. (Source: Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.)
(a) $\sqrt{2}$
(b) $\sqrt{11}$

James Kiss

Numerade Educator