Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan, Michael Sullivan, lll

Chapter 11

Sequences; Induction; the Binomial Theorem - all with Video Answers

Educators

Section 1

Sequences

Problem 1

For the function $f(x)=\frac{x-1}{x},$ find $f(2)$ and $f(3)$.

Julie Silva

Julie Silva

Numerade Educator

Problem 2

A function is a relation between two sets $D$ and $R$ such that each element $x$ in the first set $D$ is related to exactly one element $y$ in the second set $R$.

Kelsey Dondelinger

Kelsey Dondelinger

Numerade Educator

Problem 3

$\mathrm{A}(\mathrm{n})$ _________ is a function whose domain is the set of positive integers.

Julie Silva

Numerade Educator

Problem 4

The notation $a_{5}$ represents the fifth term of a sequence.

Sheryl Ezze

Numerade Educator

Problem 5

If $n \geq 0$ is an integer, then $n !=$ ____________ when $n \geq 2$.

Anurag Kumar

Numerade Educator

Problem 6

The sequence $a_{1}=5, a_{n}=3 a_{n-1}$ is an example of a ___________ sequence.

Sheryl Ezze

Numerade Educator

Problem 7

The notation $a_{1}+a_{2}+a_{3}+\cdots+a_{n}=\sum_{k=1}^{n} a_{k}$ is an example of _____________ notation.

Julie Silva

Numerade Educator

Problem 8

TRUE OR FALSE $\sum_{k=1}^{n} k=1+2+3+\cdots+n=\frac{n(n+1)}{2}$

Anurag Kumar

Numerade Educator

Problem 9

Evaluate each factorial expression.
$10 !$

Julie Silva

Numerade Educator

Problem 10

Evaluate each factorial expression.
. $9 !$

Julie Silva

Numerade Educator

Problem 11

Evaluate each factorial expression.
$\frac{9 !}{6 !}$

Julie Silva

Numerade Educator

Problem 12

Evaluate each factorial expression.
$\frac{12 !}{10 !}$

Julie Silva

Numerade Educator

Problem 13

Evaluate each factorial expression.
$\frac{3 ! 7 !}{4 !}$

Julie Silva

Numerade Educator

Problem 14

Evaluate each factorial expression.
$\frac{5 ! 8 !}{3 !}$

Julie Silva

Numerade Educator

Problem 15

Write down the first five terms of each sequence.
$\left\{s_{n}\right\}=\{n\}$

Julie Silva

Numerade Educator

Problem 16

Write down the first five terms of each sequence.
$\left\{s_{n}\right\}=\left\{n^{2}+1\right\}$

Julie Silva

Numerade Educator

Problem 17

Write down the first five terms of each sequence.
$\left\{a_{n}\right\}=\left\{\frac{n}{n+2}\right\}$

Julie Silva

Numerade Educator

Problem 18

Write down the first five terms of each sequence.
$\left\{b_{n}\right\}=\left\{\frac{2 n+1}{2 n}\right\}$

Julie Silva

Numerade Educator

Problem 19

Write down the first five terms of each sequence.
$\left\{c_{n}\right\}=\left\{(-1)^{n+1} n^{2}\right\}$

Julie Silva

Numerade Educator

Problem 20

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

Julie Silva

Numerade Educator

Problem 21

Write down the first five terms of each sequence.
$\left\{s_{n}\right\}=\left\{\frac{2^{n}}{3^{n}+1}\right\}$

Julie Silva

Numerade Educator

Problem 22

Write down the first five terms of each sequence.
$\left\{s_{n}\right\}=\left\{\left(\frac{4}{3}\right)^{n}\right\}$

Julie Silva

Numerade Educator

Problem 23

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

Julie Silva

Numerade Educator

Problem 24

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

Julie Silva

Numerade Educator

Problem 25

Write down the first five terms of each sequence.
$\left\{b_{n}\right\}=\left\{\frac{n}{e^{n}}\right\}$

Julie Silva

Numerade Educator

Problem 26

Write down the first five terms of each sequence.
$\left\{c_{n}\right\}=\left\{\frac{n^{2}}{2^{n}}\right\}$

Julie Silva

Numerade Educator

Problem 27

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$

Sheryl Ezze

Numerade Educator

Problem 28

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \cdots$

Sheryl Ezze

Numerade Educator

Problem 29

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$

Sheryl Ezze

Numerade Educator

Problem 30

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots$

Sheryl Ezze

Numerade Educator

Problem 31

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$1,-1,1,-1,1,-1, \ldots$

Sheryl Ezze

Numerade Educator

Problem 32

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \ldots$

Lauren Shelton

Numerade Educator

Problem 33

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$1,-2,3,-4,5,-6, \ldots$

Sheryl Ezze

Numerade Educator

Problem 34

Write down the nth term of a sequence $\left\{a_{n}\right\}$ suggested by the pattern.
$2,-4,6,-8,10, \ldots$

Sheryl Ezze

Numerade Educator

Problem 35

A sequence is defined recursively. Write down the first five terms.
$a_{1}=2 ; \quad a_{n}=3+a_{n-1}$

Julie Silva

Numerade Educator

Problem 36

A sequence is defined recursively. Write down the first five terms.
$a_{1}=3 ; \quad a_{n}=4-a_{n-1}$

Julie Silva

Numerade Educator

Problem 37

A sequence is defined recursively. Write down the first five terms.
$a_{1}=-2 ; \quad a_{n}=n+a_{n-1}$

Julie Silva

Numerade Educator

Problem 38

A sequence is defined recursively. Write down the first five terms.
$a_{1}=1 ; \quad a_{n}=n-a_{n-1}$

Julie Silva

Numerade Educator

Problem 39

A sequence is defined recursively. Write down the first five terms.
$a_{1}=5 ; \quad a_{n}=2 a_{n-1}$

Julie Silva

Numerade Educator

Problem 40

A sequence is defined recursively. Write down the first five terms.
$a_{1}=2 ; \quad a_{n}=-a_{n-1}$

Julie Silva

Numerade Educator

Problem 41

A sequence is defined recursively. Write down the first five terms.
$a_{1}=3 ; \quad a_{n}=\frac{a_{n-1}}{n}$

Julie Silva

Numerade Educator

Problem 42

A sequence is defined recursively. Write down the first five terms.
$a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}$

Julie Silva

Numerade Educator

Problem 43

A sequence is defined recursively. Write down the first five terms.
$a_{1}=1 ; \quad a_{2}=2 ; \quad a_{n}=a_{n-1} \cdot a_{n-2}$

Julie Silva

Numerade Educator

Problem 44

A sequence is defined recursively. Write down the first five terms.
$a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}$

Julie Silva

Numerade Educator

Problem 45

A sequence is defined recursively. Write down the first five terms.
$a_{1}=A ; \quad a_{n}=a_{n-1}+d$

Julie Silva

Numerade Educator

Problem 46

A sequence is defined recursively. Write down the first five terms.
$a_{1}=A ; \quad a_{n}=r a_{n-1}, \quad r \neq 0$

Julie Silva

Numerade Educator

Problem 47

A sequence is defined recursively. Write down the first five terms.
$a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{2+a_{n-1}}$

Julie Silva

Numerade Educator

Problem 48

A sequence is defined recursively. Write down the first five terms.
$a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{\frac{a_{n-1}}{2}}$

Julie Silva

Numerade Educator

Problem 49

Write out each sum.
$\sum_{k=1}^{n}(k+2)$

Julie Silva

Numerade Educator

Problem 50

Write out each sum.
$\sum_{k=1}^{n}(2 k+1)$

Julie Silva

Numerade Educator

Problem 51

Write out each sum.
$\sum_{k=1}^{n} \frac{k^{2}}{2}$

Julie Silva

Numerade Educator

Problem 52

Write out each sum.
$\sum_{k=1}^{n}(k+1)^{2}$

Julie Silva

Numerade Educator

Problem 53

Write out each sum.
$\sum_{k=0}^{n} \frac{1}{3^{k}}$

Julie Silva

Numerade Educator

Problem 54

Write out each sum.
$\sum_{k=0}^{n}\left(\frac{3}{2}\right)^{k}$

Julie Silva

Numerade Educator

Problem 55

Write out each sum.
$\sum_{k=0}^{n-1} \frac{1}{3^{k+1}}$

Julie Silva

Numerade Educator

Problem 56

Write out each sum.
$\sum_{k=0}^{n-1}(2 k+1)$

Julie Silva

Numerade Educator

Problem 57

Write out each sum.
$\sum_{k=2}^{n}(-1)^{k} \ln k$

Julie Silva

Numerade Educator

Problem 58

Write out each sum.
$\sum_{k=3}^{n}(-1)^{k+1} 2^{k}$

Julie Silva

Numerade Educator

Problem 59

Express each sum using summation notation.
. $1+2+3+\cdots+20$

Julie Silva

Numerade Educator

Problem 60

Express each sum using summation notation.
$1^{3}+2^{3}+3^{3}+\cdots+8^{3}$

Julie Silva

Numerade Educator

Problem 61

Express each sum using summation notation.
$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{13}{13+1}$

Julie Silva

Numerade Educator

Problem 62

Express each sum using summation notation.
$1+3+5+7+\cdots+[2(12)-1]$

Julie Silva

Numerade Educator

Problem 63

Express each sum using summation notation.
$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots+(-1)^{6}\left(\frac{1}{3^{6}}\right)$

Julie Silva

Numerade Educator

Problem 64

Express each sum using summation notation.
$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\cdots+(-1)^{12}\left(\frac{2}{3}\right)^{11}$

Julie Silva

Numerade Educator

Problem 65

Express each sum using summation notation.
$\mathbf{} \cdot 3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\cdots+\frac{3^{n}}{n}$

Julie Silva

Numerade Educator

Problem 66

Express each sum using summation notation.
$\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\cdots+\frac{n}{e^{n}}$

Julie Silva

Numerade Educator

Problem 67

Express each sum using summation notation.
$a+(a+d)+(a+2 d)+\cdots+(a+n d)$

Julie Silva

Numerade Educator

Problem 68

Express each sum using summation notation.
$a+a r+a r^{2}+\cdots+a r^{n-1}$

Julie Silva

Numerade Educator

Problem 69

Find the sum of each sequence.
$\sum_{k=1}^{40} 5$

Julie Silva

Numerade Educator

Problem 70

Find the sum of each sequence.
$\sum_{i=1}^{50} 8$

Sheryl Ezze

Numerade Educator

Problem 71

Find the sum of each sequence.
$\sum_{k=1}^{40} k$

Julie Silva

Numerade Educator

Problem 72

Find the sum of each sequence.
$\sum_{k=1}^{24}(-k)$

Julie Silva

Numerade Educator

Problem 73

Find the sum of each sequence.
$\sum_{k=1}^{20}(5 k+3)$

Julie Silva

Numerade Educator

Problem 74

Find the sum of each sequence.
$\sum_{k=1}^{26}(3 k-7)$

Julie Silva

Numerade Educator

Problem 75

Find the sum of each sequence.
$\sum_{k=1}^{16}\left(k^{2}+4\right)$

Julie Silva

Numerade Educator

Problem 76

Find the sum of each sequence.
$\sum_{k=0}^{14}\left(k^{2}-4\right)$

Julie Silva

Numerade Educator

Problem 77

Find the sum of each sequence.
$\sum_{k=10}^{60}(2 k)$

Julie Silva

Numerade Educator

Problem 78

Find the sum of each sequence.
$\sum_{k=8}^{40}(-3 k)$

Julie Silva

Numerade Educator

Problem 79

Find the sum of each sequence.
$\sum_{k=5}^{20} k^{3}$

Julie Silva

Numerade Educator

Problem 80

Find the sum of each sequence.
$\sum_{k=4}^{24} k^{3}$

Julie Silva

Numerade Educator

Problem 81

John has a balance of $$\$ 3000$$ on his Discover card, which charges $$1 \%$$ interest per month on any unpaid balance. John can afford to pay $$\$ 100$$ toward the balance each month. His balance each month after making a $$\$ 100$$ payment is given by the recursively defined sequence
$$B_{0}=\$ 3000 \quad B_{n}=1.01 B_{n-1}-100$$
Determine John's balance after making the first payment. That is, determine $B_{1}$.

Sheryl Ezze

Numerade Educator

Problem 82

Trout Population A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. It is also known that the trout population is growing at a rate of $3 \%$ per month. The size of the population after $n$ months is given by the recursively defined sequence
$$p_{0}=2000 \quad p_{n}=1.03 p_{n-1}+20$$
How many trout are in the pond after 2 months? That is, what is $p_{2} ?$

Julie Silva

Numerade Educator

Problem 83

Car Loans Phil bought a car by taking out a loan for $$\$ 18,500$$ at $$0.5 \%$$ interest per month. Phil's normal monthly payment is $$\$ 434.47$$ per month, but he decides that he can afford to pay $$\$ 100$$ extra toward the balance each month. His balance each month is given by the recursively defined
sequence
$$B_{0}=\$ 18,500 \quad B_{n}=1.005 B_{n-1}-534.47$$
Determine Phil's balance after making the first payment. That is, determine $B_{1}$.

Julie Silva

Numerade Educator

Problem 84

The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that $10 \%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution-control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursively defined sequence
$$p_{0}=250 \quad p_{n}=0.9 p_{n-1}+15$$
Determine the amount of pollutant in the lake after 2 years. That is, determine $p_{2}$.

Julie Silva

Numerade Educator

Problem 85

A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months?

Bryan Lynn

Numerade Educator

Problem 86

Fibonacci Sequence Let
$$u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$
define the $n$ th term of a sequence.
(a) Show that $u_{1}=1$ and $u_{2}=1$.
(b) Show that $u_{n+2}=u_{n+1}+u_{n}$.
(c) Draw the conclusion that $\left\{u_{n}\right\}$ is a Fibonacci sequence.

Anurag Kumar

Numerade Educator

Problem 87

Divide the triangular array shown (called Pascal's triangle) using diagonal lines as indicated. Find the sum of the numbers in each diagonal row. Do you recognize this sequence?

Ziya Ogron

Numerade Educator

Problem 88

Use the result of Problem 86 to do the following problems.
(a) Write the first 11 terms of the Fibonacci sequence.
(b) Write the first 10 terms of the ratio $\frac{u_{n+1}}{u_{n}}$.
(c) As $n$ gets large, what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the façade of the Parthenon was constructed using the golden ratio.
(d) Write down the first 10 terms of the ratio $\frac{u_{n}}{u_{n+1}}$.
(e) As $n$ gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.

Anurag Kumar

Numerade Educator

Problem 89

$f(x)=e^{x}$ In calculus, it can be shown that
$$f(x)=e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}$$
We can approximate the value of $f(x)=e^{x}$ for any $x$ using the following sum
$$f(x)=e^{x} \approx \sum_{k=0}^{n} \frac{x^{k}}{k !}$$
for some $n$.
(a) Approximate $f(1.3)$ with $n=4$
(b) Approximate $f(1.3)$ with $n=7$.
(c) Use a calculator to approximate $f(1.3)$.
(d) Using trial and error, along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(1.3)$ correct to eight decimal places.

Anurag Kumar

Numerade Educator

Problem 90

$f(x)=e^{x}$ Refer to Problem $89 .$
(a) Approximate $f(-2.4)$ with $n=3$.
(b) Approximate $f(-2.4)$ with $n=6$.
(c) Use a calculator to approximate $f(-2.4)$.
(d) Using trial and error, along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(-2.4)$ correct to eight decimal places.

Anurag Kumar

Numerade Educator

Problem 91

In $1772,$ Johann Bode published the following formula for predicting the mean distances, in astronomical units $(\mathrm{AU}),$ of the planets from the sun:
$$a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2}, n \geq 2$$
where $n$ is the number of the planet from the sun.
(a) Determine the first eight terms of this sequence.
(b) At the time of Bode's publication, the known planets were Mercury $(0.39 \mathrm{AU}),$ Venus $(0.72 \mathrm{AU}),$ Earth $(1 \mathrm{AU}),$ Mars $(1.52 \mathrm{AU}),$ Jupiter $(5.20 \mathrm{AU}),$ and Saturn $(9.54 \mathrm{AU})$. How do the actual distances compare to the terms of the sequence?
(c) The planet Uranus was discovered in 1781 and the asteroid Ceres was discovered in 1801 . The mean orbital distances from the sun to Uranus and Ceres" are 19.2 $\mathrm{AU}$ and $2.77 \mathrm{AU},$ respectively. How well do these values fit within the sequence?
(d) Determine the ninth and tenth terms of Bode's sequence.
(e) The planets Neptune and Pluto $^{\circ}$ were discovered in 1846 and 1930 , respectively. Their mean orbital distances from the sun are $30.07 \mathrm{AU}$ and $39.44 \mathrm{AU},$ respectively. How do these actual distances compare to the terms of the sequence?
(f) On July $29,2005,$ NASA announced the discovery of a dwarf planet $^{*}(n=11),$ which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Anurag Kumar

Numerade Educator

Problem 92

Show that
$$1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2}$$

Sheryl Ezze

Numerade Educator

Problem 93

Computing Square Roots A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.
$\sqrt{5}$

Anurag Kumar

Numerade Educator

Problem 94

Computing Square Roots A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.
$\sqrt{8}$

Anurag Kumar

Numerade Educator

Problem 95

Computing Square Roots A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.
$\sqrt{21}$

Anurag Kumar

Numerade Educator

Problem 96

Computing Square Roots A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$a_{0}=k \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_{5} .$ Compare this result to the value provided by your calculator.
$\sqrt{89}$

Anurag Kumar

Numerade Educator

Problem 97

Investigate various applications that lead to a Fibonacci sequence, such as in art, architecture, or financial markets. Write an essay on these applications.

Julie Silva

Numerade Educator

Problem 98

If $$\$ 2500$$ is invested at $3 \%$ compounded monthly, find the amount that results after a period of 2 years.

Julie Silva

Numerade Educator

Problem 99

Write the complex number $-1-i$ in polar form. Express the argument in degrees.

Lauren Shelton

Numerade Educator

Problem 100

For $\mathbf{v}=2 \mathbf{i}-\mathbf{j}+3 \mathbf{k}$ and $\mathbf{w}=\mathbf{i}+2 \mathbf{j}-\mathbf{k},$ find the cross product $\mathbf{v} \times \mathbf{w} .$

Vikash Ranjan

Numerade Educator

Problem 101

Find an equation of the parabola with vertex (-3,4) and focus (1,4) .

Julie Silva

Numerade Educator