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Discrete Mathematics and Its Applications

Kenneth H. Rosen

Chapter 4

Induction and Recursion $\qquad$ - all with Video Answers

Educators

Section 1

Mathematical Induction. $\qquad$

02:07

Problem 1

There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:41

Problem 2

Suppose that you know that a golfer plays the first hole of a golf course with an infinite number of holes and that if this golfer plays one hole, then the golfer goes on to play the next hole. Prove that this golfer plays every hole on the course.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
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Problem 3

Let $P(n)$ be the statement that $1^2+2^2+\cdots+n^2=$ $n(n+1)(2 n+1) / 6$ for the positive integer $n$.
a) What is the statement $P(1)$ ?
b) Show that $P(1)$ is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step.
f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Chloe Frechette
Chloe Frechette
Numerade Educator
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Problem 4

Let $P(n)$ be the statement that $1^3+2^3+\cdots+n^3=$ $(n(n+1) / 2)^2$ for the positive integer $n$.
a) What is the statement $P(1)$ ?
b) Show that $P(1)$ is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step.
f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Chloe Frechette
Chloe Frechette
Numerade Educator
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Problem 5

Prove that $1^2+3^2+5^2+\cdots+(2 n+1)^2=(n+1)$ $(2 n+1)(2 n+3) / 3$ whenever $n$ is a nonnegative integer.

Chloe Frechette
Chloe Frechette
Numerade Educator
03:07

Problem 6

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$ whenever $n$ is a positive integer.

Julian Wong
Julian Wong
Numerade Educator
03:09

Problem 7

Prove that $3+3 \cdot 5+3 \cdot 5^2+\cdots+3 \cdot 5^n=3\left(5^{n+1}-1\right) / 4$ whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
04:40

Problem 8

Prove that $2-2 \cdot 7+2 \cdot 7^2-\cdots+2(-7)^n=(1-$ $\left.(-7)^{n+1}\right) / 4$ whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
09:41

Problem 9

a) Find a formula for the sum of the first $n$ even positive integers.
b) Prove the formula that you conjectured in part (a).

Raphael Tinoco
Raphael Tinoco
Numerade Educator
02:37

Problem 10

a) Find a formula for
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}
$$
by examining the values of this expression for small values of $n$.
b) Prove the formula you conjectured in part (a).

Farnood Ensan
Farnood Ensan
Numerade Educator
02:27

Problem 11

a) Find a formula for
$$
\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}
$$
by examining the values of this expression for small values of $n$.
b) Prove the formula you conjectured in part (a).

WM
William Mead
Numerade Educator
02:31

Problem 12

Prove that
$$
\sum_{j=0}^n\left(-\frac{1}{2}\right)^j=\frac{2^{n+1}+(-1)^n}{3 \cdot 2^n}
$$
whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:16

Problem 13

Prove that $1^2-2^2+3^2-\cdots+(-1)^{n-1} n^2=(-1)^{n-1}$ $n(n+1) / 2$ whenever $n$ is a positive integer.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:33

Problem 14

Prove that for every positive integer $n, \sum_{k=1}^n k 2^k=$ $(n-1) 2^{n+1}+2$.

Dushyant Barot
Dushyant Barot
Numerade Educator
02:46

Problem 15

Prove that for every positive integer $n$,
$$
1 \cdot 2+2 \cdot 3+\cdots+n(n+1)=n(n+1)(n+2) / 3 .
$$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:46

Problem 16

Prove that for every positive integer $n$,
$$
\begin{aligned}
1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+\cdots & +n(n+1)(n+2) \\
& =n(n+1)(n+2)(n+3) / 4 .
\end{aligned}
$$

Sanchit Jain
Sanchit Jain
Numerade Educator

Problem 17

Prove that $\sum_{j=1}^n j^4=n(n+1)(2 n+1)\left(3 n^2+3 n-1\right) /$ 30 whenever $n$ is a positive integer.

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04:25

Problem 18

Let $P(n)$ be the statement that $n!<n^n$, where $n$ is an integer greater than 1.
a) What is the statement $P(2)$ ?
b) Show that $P(2)$ is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step.
f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Clayton Schubring
Clayton Schubring
Numerade Educator
01:46

Problem 19

Let $P(n)$ be the statement that
$$
1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}<2-\frac{1}{n} .
$$
where $n$ is an integer greater than 1 .
a) What is the statement $P(2)$ ?
b) Show that $P(2)$ is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step.
f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Clayton Schubring
Clayton Schubring
Numerade Educator
03:02

Problem 20

Prove that $3^n<n$ ! if $n$ is an integer greater than 6 .

Clayton Schubring
Clayton Schubring
Numerade Educator
03:26

Problem 21

Prove that $2^n>n^2$ if $n$ is an integer greater than 4 .

Clayton Schubring
Clayton Schubring
Numerade Educator
02:32

Problem 22

For which nonnegative integers $n$ is $n^2 \leq n$ !? Prove your answer.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 23

For which nonnegative integers $n$ is $2 n+3 \leq 2^n$ ? Prove your answer.

Claire Rochford
Claire Rochford
Numerade Educator
01:44

Problem 24

Prove that $1 /(2 n) \leq[1 \cdot 3 \cdot 5 \cdots \cdot(2 n-1)] /(2 \cdot 4 \cdot \cdots$ $2 n$ ) whenever $n$ is a positive integer.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:11

Problem 25

Prove that if $h>-1$, then $1+n h \leq(1+h)^n$ for all nonnegative integers $n$. This is called Bernoulli's inequality.

Carson Merrill
Carson Merrill
Numerade Educator
02:57

Problem 26

Suppose that $a$ and $b$ are real numbers with $0<b<a$. Prove that if $n$ is a positive integer, then $a^n-b^n \leq$ $n a^{n-1}(a-b)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
05:51

Problem 27

Prove that for every positive integer $n$,
$$
1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1) .
$$

Linda Hand
Linda Hand
Numerade Educator
View

Problem 28

Prove that $n^2-7 n+12$ is nonnegative whenever $n$ is an integer with $n \geq 3$.

Claire Rochford
Claire Rochford
Numerade Educator
02:50

Problem 29

$H_n$ denotes the $n$th harmonic number.
Prove that $H_2 \leq 1+n$ whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:10

Problem 30

$H_n$ denotes the $n$th harmonic number.
Prove that
$$
H_1+H_2+\cdots+H_n=(n+1) H_n-n .
$$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:40

Problem 31

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that 2 divides $n^2+n$ whenever $n$ is a positive integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:51

Problem 32

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that 3 divides $n^3+2 n$ whenever $n$ is a positive integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
04:51

Problem 33

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that 5 divides $n^5-n$ whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:38

Problem 34

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that 6 divides $n^3-n$ whenever $n$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
06:31

Problem 35

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that $n^2-1$ is divisible by 8 whenever $n$ is an odd positive integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
04:44

Problem 36

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that 21 divides $4^{n+1}+5^{2 n-1}$ whenever $n$ is a positive integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
06:57

Problem 37

Use mathematical induction in Exercises 31-37 to prove divisibility facts.
Prove that if $n$ is a positive integer, then 133 divides $11^{n+1}+12^{2 n-1}$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:54

Problem 38

Prove that if $A_1, A_2, \ldots, A_n$ and $B_1, B_2, \ldots, B_n$ are sets such that $A_j \subseteq B_j$ for $j=1,2, \ldots, n$, then
$$
\bigcup_{j=1}^n A_j \subseteq \bigcup_{j=1}^n B_j .
$$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:41

Problem 39

Prove that if $A_1, A_2, \ldots, A_n$ and $B_1, B_2, \ldots, B_n$ are sets such that $A_j \subseteq B_j$ for $j=1,2, \ldots, n$, then
$$
\bigcap_{j=1}^n A_j \subseteq \bigcap_{j=1}^n B_j .
$$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator

Problem 40

Prove that if $A_1, A_2, \ldots, A_n$ and $B$ are sets, then
$$
\begin{aligned}
\left(A_1 \cap A_2 \cap\right. & \left.\cdots \cap A_n\right) \cup B \\
& =\left(A_1 \cup B\right) \cap\left(A_2 \cup B\right) \cap \cdots \cap\left(A_n \cup B\right) .
\end{aligned}
$$

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Problem 41

Prove that if $A_1, A_2, \ldots, A_n$ and $B$ are sets, then
$$
\begin{aligned}
\left(A_1 \cup A_2\right. & \left.\cup \cdots \cup A_n\right) \cap B \\
& =\left(A_1 \cap B\right) \cup\left(A_2 \cap B\right) \cup \cdots \cup\left(A_n \cap B\right) .
\end{aligned}
$$

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Problem 42

Prove that if $A_1, A_2, \ldots, A_n$ and $B$ are sets, then
$$
\begin{aligned}
\left(A_1-B\right) & \cap\left(A_2-B\right) \cap \cdots \cap\left(A_n-B\right) \\
& =\left(A_1 \cap A_2 \cap \cdots \cap A_n\right)-B .
\end{aligned}
$$

Check back soon!

Problem 43

Prove that if $A_1, A_2, \ldots, A_n$ are subsets of a universal set $U$, then
$$
\overline{\bigcup_{k=1}^n A_k}=\bigcap_{k=1}^n \overline{A_k} .
$$

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Problem 44

Prove that if $A_1, A_2, \ldots, A_n$ and $B$ are sets, then
$$
\begin{aligned}
&\left(A_1-B\right) \cup\left(A_2-B\right) \cup \cdots \cup\left(A_n-B\right) \\
&=\left(A_1 \cup A_2 \cup \cdots \cup A_n\right)-B .
\end{aligned}
$$

Check back soon!
02:21

Problem 45

Prove that a set with $n$ elements has $n(n-1) / 2$ subsets containing exactly two elements whenever $n$ is an integer greater than or equal to 2 .

Nick Johnson
Nick Johnson
Numerade Educator
02:21

Problem 46

Prove that a set with $n$ elements has $n(n-1)(n-2) / 6$ subsets containing exactly three elements whenever $n$ is an integer greater than or equal to 3 .

Nick Johnson
Nick Johnson
Numerade Educator
01:17

Problem 47

What is wrong with this "proof" that all horses are the same color?
Let $P(n)$ be the proposition that all the horses in a set of $n$ horses are the same color.
Basis Step: Clearly, $P(1)$ is true.
Inductive Step: Assume that $P(k)$ is true, so that all the horses in any set of $k$ horses are the same color.
Consider any $k+1$ horses; number these as horses $1,2,3, \ldots, k, k+1$. Now the first $k$ of these horses all must have the same color, and the last $k$ of these must also have the same color. Because the set of the first $k$ horses and the set of the last $k$ horses overlap, all $k+1$ must be the same color. This shows that $P(k+1)$ is true and finishes the proof by induction.

Nick Johnson
Nick Johnson
Numerade Educator
01:59

Problem 48

What is wrong with this "proof"?
"Theorem" For every positive integer $n, \sum_{i=1}^n i=$ $\left(n+\frac{1}{2}\right)^2 / 2$.
Basis Step: The formula is true for $n=1$.
Inductive Step: Suppose that $\sum_{i=1}^n i=\left(n+\frac{1}{2}\right)^2 / 2$. Then $\sum_{i=1}^{n+1} i=\left(\sum_{i=1}^n i\right)+(n+1)$. By the inductive hypothesis, $\sum_{i=1}^{n+1} i=\left(n+\frac{1}{2}\right)^2 / 2+n+1=\left(n^2+n+\right.$ $\left.\frac{1}{4}\right) / 2+n+1=\left(n^2+3 n+\frac{9}{4}\right) / 2=\left(n+\frac{3}{2}\right)^2 / 2=$ $\left[(n+1)+\frac{1}{2}\right]^2 / 2$, completing the inductive step.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:39

Problem 49

What is wrong with this "proof"?
"Theorem" For every positive integer $n$, if $x$ and $y$ are positive integers with $\max (x, y)=n$, then $x=y$.
Basis Step: Suppose that $n=1$. If $\max (x, y)=1$ and $x$ and $y$ are positive integers, we have $x=1$ and $y=1$.
Inductive Step: Let $k$ be a positive integer. Assume that whenever $\max (x, y)=k$ and $x$ and $y$ are positive integers, then $x=y$. Now let $\max (x, y)=k+1$, where $x$ and $y$ are positive integers. Then $\max (x-1, y-1)=k$, so by the inductive hypothesis, $x-1=y-1$. It follows that $x=y$, completing the inductive step.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
11:23

Problem 50

Use mathematical induction to show that given a set of $n+1$ positive integers, none exceeding $2 n$, there is at least one integer in this set that divides another integer in the set.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:21

Problem 51

A knight on a chessboard can move one space horizontally (in either direction) and two spaces vertically (in either direction) or two spaces horizontally (in either direction) and one space vertically (in either direction). Suppose that we have an infinite chessboard, made up of all squares $(m, n)$ where $m$ and $n$ are nonnegative integers. Use mathematical induction to show that a knight starting at $(0,0)$ can visit every square using a finite sequence of moves.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:46

Problem 52

Suppose that
$$
\mathbf{A}=\left[\begin{array}{ll}
a & 0 \\
0 & b
\end{array}\right] .
$$
where $a$ and $b$ are real numbers. Show that
$$
\mathbf{A}^n=\left[\begin{array}{cc}
a^n & 0 \\
0 & b^n
\end{array}\right]
$$
for every positive integer $n$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:31

Problem 53

(Requires calculus) Use mathematical induction to prove that the derivative of $f(x)=x^n$ equals $n x^{n-1}$ whenever $n$ is a positive integer. (For the inductive step, use the product rule for derivatives.)

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:43

Problem 54

Suppose that $\mathbf{A}$ and $\mathbf{B}$ are square matrices with the property $\mathbf{A B}=\mathbf{B A}$. Show that $\mathbf{A B}^n=\mathbf{B}^n \mathbf{A}$ for every positive integer $n$.

Nick Johnson
Nick Johnson
Numerade Educator
02:51

Problem 55

Suppose that $m$ is a positive integer. Use mathematical induction to prove that if $a$ and $b$ are integers with $a \equiv b$ $(\bmod m)$, then $a^k \equiv b^k(\bmod m)$ whenever $k$ is a nonnegative integer.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:03

Problem 56

Use mathematical induction to show that $\neg\left(p_1 \vee p_2 \vee\right.$ $\cdots \vee p_n$ ) is equivalent to $\neg p_1 \wedge \neg p_2 \wedge \cdots \wedge \neg p_n$ whenever $p_1, p_2, \ldots, p_n$ are propositions.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:17

Problem 57

Show that
$$
\begin{aligned}
{\left[\left(p_1 \rightarrow p_2\right) \wedge\left(p_2 \rightarrow p_3\right) \wedge \cdots \wedge\left(p_{n-1} \rightarrow p_n\right)\right] } \\
\rightarrow\left[\left(p_1 \wedge p_2 \wedge \cdots \wedge p_{n-1}\right) \rightarrow p_n\right]
\end{aligned}
$$
is a tautology whenever $p_1, p_2, \ldots, p_n$ are propositions, where $n \geq 2$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
06:42

Problem 58

Show that $n$ lines separate the plane into $\left(n^2+n+2\right) / 2$ regions if no two of these lines are parallel and no three pass through a common point.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
05:29

Problem 59

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. The arithmetic mean of these numbers is defined by
$$
A=\left(a_1+a_2+\cdots+a_n\right) / n,
$$
and the geometric mean of these numbers is defined by
$$
G=\left(a_1 a_2 \cdots a_n\right)^{1 / n} .
$$

Use mathematical induction to prove that $A \geq G$.

Carson Merrill
Carson Merrill
Numerade Educator
03:47

Problem 60

Use mathematical induction to prove Lemma 2 of Section 3.6 , which states that if $p$ is a prime and $p \mid a_1 a_2 \cdots a_n$, where $a_i$ is an integer for $i=1,2,3, \ldots, n$, then $p \mid a_i$ for some integer $i$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:17

Problem 61

Show that if $n$ is a positive integer, then
$$
\sum_{\left.\mid a a_1, \ldots, a_k\right) \subseteq(1.2, \ldots, n \mid} \frac{1}{a_1 a_2 \cdots a_k}=n .
$$
(Here the sum is over all nonempty subsets of the set of the $n$ smallest positive integers.)

Carson Merrill
Carson Merrill
Numerade Educator
04:52

Problem 62

Use the well-ordering property to show that the following form of mathematical induction is a valid method to prove that $P(n)$ is true for all positive integers $n$.
Basis Step: $P(1)$ and $P(2)$ are true.
Inductive Step: For each positive integer $k$, if $P(k)$ and $P(k+1)$ are both true, then $P(k+2)$ is true.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
08:42

Problem 63

Show that if $A_1, A_2, \ldots, A_n$ are sets where $n \geq 2$, and for all pairs of integers $i$ and $j$ with $1 \leq i<j \leq n$ either $A_i$ is a subset of of $A_j$ or $A_j$ is a subset of $A_j$, then there is an integer $i, 1 \leq i \leq n$ such that $A_i$ is a subset of $A_j$ for all integers $j$ with $1 \leq j \leq n$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:54

Problem 64

A guest at a party is a celebrity if this person is known by every other guest, but knows none of them. There is at most one celebrity at a party, for if there were two, they would know each other. A particular party may have no celebrity. Your assignment is to find the celebrity, if one exists, at a party, by asking only one type of questionasking a guest whether they know a second guest. Everyone must answer your questions truthfully. That is, if Alice and Bob are two people at the party, you can ask Alice whether she knows Bob; she must answer correctly. Use mathematical induction to show that if there are $n$ people at the party, then you can find the celebrity, if there is one, with $3(n-1)$ questions.

Carson Merrill
Carson Merrill
Numerade Educator
01:41

Problem 65

Find $G(1), G(2), G(3)$, and $G(4)$.

Julie Silva
Julie Silva
Numerade Educator
03:38

Problem 66

Use mathematical induction to prove that $G(n) \leq 2 n-4$ for $n \geq 4$.

EI
Eric Icaza
Numerade Educator

Problem 67

Prove that $G(n)=2 n-4$ for $n \geq 4$.

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05:47

Problem 68

Show that it is possible to arrange the numbers $1,2, \ldots, n$ in a row so that the average of any two of these numbers never appears between them. [Hint; Show that it suffices to prove this fact when $n$ is a power of 2 . Then use mathematical induction to prove the result when $n$ is a power of 2.]

Carson Merrill
Carson Merrill
Numerade Educator
12:39

Problem 69

Show that if $I_1, I_2, \ldots, I_n$ is a collection of open intervals on the real number line, $n \geq 2$, and every pair of these intervals has a nonempty intersection, that is, $I_i \cap I_j \neq \emptyset$ whenever $1 \leq i \leq n$ and $1 \leq j \leq n$, then the intersection of all these sets is nonempty, that is, $I_1 \cap I_2 \cap \cdots \cap I_n \neq$ $\emptyset$. (Recall that an open interval is the set of real numbers $x$ with $a<x<b$, where $a$ and $b$ are real numbers with $a<b$.)

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
03:27

Problem 70

Suppose that we want to prove that
$$
\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n}}
$$
for all positive integers $n$.
a) Show that if we try to prove this inequality using mathematical induction, the basis step works, but the inductive step fails.
b) Show that mathematical induction can be used to prove the stronger inequality
$$
\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n+1}}
$$
for all integers greater than 1 , which, together with a verification for the case where $n=1$, establishes the weaker inequality we originally tried to prove using mathematical induction.

Nick Johnson
Nick Johnson
Numerade Educator
03:18

Problem 71

Let $n$ be an even positive integer. Show that when $n$ people stand in a yard at mutually distinct distances and each person throws a pie at their nearest neighbor, it is possible that everyone is hit by a pie.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
00:40

Problem 72

Construct a tiling using right triominoes of the $4 \times 4$ checkerboard with the square in the upper left corner removed.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
00:30

Problem 73

Construct a tiling using right triominoes of the $8 \times 8$ checkerboard with the square in the upper left corner removed.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
04:58

Problem 74

Prove or disprove that all checkerboards of these shapes can be completely covered using right triominoes whenever $n$ is a positive integer.
a) $3 \times 2^n$
b) $6 \times 2^n$
c) $3^n \times 3^n$
d) $6^{n 1} \times 6^n$

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator

Problem 75

Show that a three-dimensional $2^n \times 2^n \times 2^n$ checkerboard with one $1 \times 1 \times 1$ cube missing can be completely covered by $2 \times 2 \times 2$ cubes with one $1 \times 1 \times 1$ cube removed.

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Problem 76

Show that an $n \times n$ checkerboard with one square removed can be completely covered using right triominoes if $n>5, n$ is odd, and $3 Y n$.

Nick Johnson
Nick Johnson
Numerade Educator
00:29

Problem 77

Show that a $5 \times 5$ checkerboard with a corner square removed can be tiled using right triominoes.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
05:36

Problem 78

Find a $5 \times 5$ checkerboard with a square removed that cannot be tiled using right triominoes. Prove that such a tiling does not exist for this board.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
04:40

Problem 79

Use the principle of mathematical induction to show that $P(n)$ is true for $n=b, b+1, b+2, \ldots$, where $b$ is an integer, if $P(b)$ is true and the conditional statement $P(k) \rightarrow P(k+1)$ is true for all positive integers $k$ with $k \geq b$.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator