Precalculus Student Solutions Manual 5th

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

Chapter 11 Further Topics in Algebra - all with Video Answers

Educators

Section 1

Sequences and Series

00:57

Problem 1

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

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00:59

Problem 2

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

Rakesh Kumar Sharma

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01:15

Problem 3

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

Rakesh Kumar Sharma

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01:17

Problem 4

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

Rakesh Kumar Sharma

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01:22

Problem 5

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

Rakesh Kumar Sharma

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00:45

Problem 6

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

Maninder Singh

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01:06

Problem 7

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

Rakesh Kumar Sharma

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01:26

Problem 8

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

Rakesh Kumar Sharma

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01:59

Problem 9

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

Rakesh Kumar Sharma

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01:24

Problem 10

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

Rakesh Kumar Sharma

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01:55

Problem 11

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

Rakesh Kumar Sharma

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02:17

Problem 12

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

Rakesh Kumar Sharma

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01:16

Problem 13

Your friend does not understand what is meant by the $n$ th term, or general term, of a sequence. How would you explain this idea?

Ankit Gupta

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01:47

Problem 14

How are sequences related to functions?

Ankit Gupta

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00:40

Problem 15

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

Ankit Gupta

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00:52

Problem 16

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

Rakesh Kumar Sharma

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00:33

Problem 17

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

Rakesh Kumar Sharma

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00:39

Problem 18

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

Rakesh Kumar Sharma

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00:37

Problem 19

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

Rakesh Kumar Sharma

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00:29

Problem 20

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

Rakesh Kumar Sharma

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01:28

Problem 21

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

Rakesh Kumar Sharma

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01:31

Problem 22

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

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01:23

Problem 23

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

Ankit Gupta

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01:22

Problem 24

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

Ankit Gupta

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01:33

Problem 25

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.)

Ankit Gupta

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01:06

Problem 26

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

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01:21

Problem 27

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

Ankit Gupta

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01:36

Problem 28

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

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01:13

Problem 29

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

Ankit Gupta

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01:03

Problem 30

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

Ankit Gupta

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01:14

Problem 31

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

Rakesh Kumar Sharma

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00:51

Problem 32

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

Amy Jiang

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00:52

Problem 33

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

Ankit Gupta

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01:25

Problem 34

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

Rakesh Kumar Sharma

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01:00

Problem 35

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

Ankit Gupta

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01:48

Problem 36

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

Ankit Gupta

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01:28

Problem 37

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

Ankit Gupta

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01:41

Problem 38

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

Ankit Gupta

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02:29

Problem 39

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

Chris Wojturski

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01:04

Problem 40

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

Rakesh Kumar Sharma

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01:17

Problem 41

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

Ankit Gupta

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01:04

Problem 42

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

Ankit Gupta

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00:57

Problem 43

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

Ankit Gupta

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01:08

Problem 44

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

Ankit Gupta

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00:37

Problem 45

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

Rakesh Kumar Sharma

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01:47

Problem 46

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

Rakesh Kumar Sharma

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00:42

Problem 47

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

Rakesh Kumar Sharma

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00:39

Problem 48

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

Rakesh Kumar Sharma

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00:44

Problem 49

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

Rakesh Kumar Sharma

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00:44

Problem 50

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

Rakesh Kumar Sharma

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00:37

Problem 51

Write the terms for each series. 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

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00:42

Problem 52

Write the terms for each series. 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

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01:19

Problem 53

Write the terms for each series. 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

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01:06

Problem 54

Write the terms for each series. 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

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01:31

Problem 55

Write the terms for each series. 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

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01:00

Problem 56

Write the terms for each series. 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

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02:19

Problem 57

Write the terms for each series. 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

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02:14

Problem 58

Write the terms for each series. 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

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02:53

Problem 59

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

Rakesh Kumar Sharma

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01:05

Problem 60

Explain how factoring can make the work in Exercises $11,12,$ and 59 easier.

Rakesh Kumar Sharma

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01:52

Problem 61

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 =ach function. Evaluate the sum.
$$f(x)=4 x-7$$

Rakesh Kumar Sharma

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01:49

Problem 62

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 =ach function. Evaluate the sum.
$$f(x)=6+2 x$$

Rakesh Kumar Sharma

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01:31

Problem 63

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 =ach function. Evaluate the sum.
$$f(x)=2 x^{2}$$

Rakesh Kumar Sharma

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01:23

Problem 64

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 =ach function. Evaluate the sum.
$$f(x)=x^{2}-1$$

Rakesh Kumar Sharma

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02:43

Problem 65

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 =ach function. Evaluate the sum.
$$f(x)=\frac{-2}{x+1}$$

Rakesh Kumar Sharma

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02:34

Problem 66

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 =ach function. Evaluate the sum.
$$f(x)=\frac{5}{2 x-1}$$

Rakesh Kumar Sharma

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00:31

Problem 67

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

Rakesh Kumar Sharma

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00:34

Problem 68

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

Rakesh Kumar Sharma

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01:11

Problem 69

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

Rakesh Kumar Sharma

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01:48

Problem 70

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

Rakesh Kumar Sharma

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01:06

Problem 71

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

Rakesh Kumar Sharma

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00:48

Problem 72

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

Rakesh Kumar Sharma

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01:18

Problem 73

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

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01:25

Problem 74

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

Rakesh Kumar Sharma

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01:56

Problem 75

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

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01:29

Problem 76

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

Rakesh Kumar Sharma

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00:34

Problem 77

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

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00:43

Problem 78

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

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01:22

Problem 79

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

Rakesh Kumar Sharma

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01:34

Problem 80

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

Rakesh Kumar Sharma

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01:58

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 you think it converges, determine the number to which it converges.
$$a_{n}=\frac{n+4}{2 n}$$

Rakesh Kumar Sharma

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01:49

Problem 82

Rakesh Kumar Sharma

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01:48

Problem 83

Rakesh Kumar Sharma

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01:41

Problem 84

Rakesh Kumar Sharma

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03:13

Problem 85

Rakesh Kumar Sharma

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02:04

Problem 86

Rakesh Kumar Sharma

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05:20

Problem 87

(a) $a_{1}=8$ thousand per acre, $a_{2}=10.4$ thousand per acre, $a_{3}=8.528$ thousand per acre
(b) The population density oscillates above and below 9.5 thousand per acre (approximately).

Michael Cooper

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02:50

Problem 88

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

Rakesh Kumar Sharma

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09:40

Problem 89

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.

Nick Johnson

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03:20

Problem 90

Refer to Exercise 89. 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 $j \geq 1 .$ (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 as to why $K$ is called the saturation constant. Test your conjecture by changing the value of $K$ in the given formula.

Ankit Gupta

Numerade Educator

10:00

Problem 91

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

Numerade Educator

02:38

Problem 92

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 your answer to the actual decimal approximation of $\pi$.

Rakesh Kumar Sharma

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02:52

Problem 93

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

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04:51

Problem 94

The recursively defined sequence $a_{1}=k$
$$ a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right), \quad \text { if } n>1$$ 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 your result with the actual value. (Source: Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.)
(a) $\sqrt{2}$
(b) $\sqrt{11}$

James Kiss

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