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Chapter 5

Induction and Recursion

Educators


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 J.
Numerade Educator

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 J.
Numerade Educator

Problem 3

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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 a proof that $P(n)$ is true for all positive integers $n .$
c) What is the inductive hypothesis of a proof that $P(n)$ is true for all positive integers $n ?$
d) What do you need to prove in the inductive step of a proof that $P(n)$ is true for all positive integers $n ?$
e) Complete the inductive step of a proof that $P(n)$ is true for all positive integers $n$ , identifying where you use the inductive hypothesis.
f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Prathan J.
Numerade Educator

Problem 4

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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 of $P(n)$ for all positive integers $n .$
c) What is the inductive hypothesis of a proof that $P(n)$ is true for all positive integers $n$ ?
d) What do you need to prove in the inductive step of a proof that $P(n)$ is true all positive integers $n ?$
e) Complete the inductive step of a proof that $P(n)$ is true for all positive integers $n,$ identifying where you use the inductive hypothesis.
f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Prathan J.
Numerade Educator

Problem 5

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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.

Prathan J.
Numerade Educator

Problem 6

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Prove that $1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !=(n+1) !-1$ whenever $n$ is a positive integer.

Prathan J.
Numerade Educator

Problem 7

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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 J.
Numerade Educator

Problem 8

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Prove that $2-2 \cdot 7+2 \cdot 7^{2}-\cdots+2(-7)^{n}=(1-$ $(-7)^{n+1} ) / 4$ whenever $n$ is a nonnegative integer.

Prathan J.
Numerade Educator

Problem 9

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
a) Find a formula for the sum of the first $n$ even positive integers.
b) Prove the formula that you conjectured in part (a).

Prathan J.
Numerade Educator

Problem 10

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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).

Prathan J.
Numerade Educator

Problem 11

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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).

Prathan J.
Numerade Educator

Problem 12

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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 J.
Numerade Educator

Problem 13

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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.

Prathan J.
Numerade Educator

Problem 14

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Prove that for every positive integer $n, \sum_{k=1}^{n} k 2^{k}=$ $(n-1) 2^{n+1}+2$.

Prathan J.
Numerade Educator

Problem 15

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Prove that for every positive integer $n$
$$
1 \cdot 2+2 \cdot 3+\cdots+n(n+1)=n(n+1)(n+2) / 3
$$

Prathan J.
Numerade Educator

Problem 16

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Prove that for every positive integer $n$
$\begin{aligned} 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+\cdots &+n(n+1)(n+2) \\ &=n(n+1)(n+2)(n+3) / 4 \end{aligned}$

Prathan J.
Numerade Educator

Problem 17

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
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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Problem 18

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
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 a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than $1 .$
c) What is the inductive hypothesis of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1 ?
d) What do you need to prove in the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$
e) Complete the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than $1 .$
f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Prathan J.
Numerade Educator

Problem 19

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
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 a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1 .
c) What is the inductive hypothesis of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$
d) What do you need to prove in the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$
e) Complete the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than $1 .$
f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Prathan J.
Numerade Educator

Problem 20

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that $3^{n}<n !$ if $n$ is an integer greater than 6

Prathan J.
Numerade Educator

Problem 21

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that $2^{n}>n^{2}$ if $n$ is an integer greater than 4

Prathan J.
Numerade Educator

Problem 22

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
For which nonnegative integers $n$ is $n^{2} \leq n ! ?$ Prove your answer.

Prathan J.
Numerade Educator

Problem 23

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
For which nonnegative integers $n$ is $2 n+3 \leq 2^{n} ?$ Prove your answer.

Prathan J.
Numerade Educator

Problem 24

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that $1 /(2 n) \leq[1 \cdot 3 \cdot 5 \cdots \cdot(2 n-1)] /(2 \cdot 4 \cdots .$ 2$n$ ) whenever $n$ is a positive integer.

Prathan J.
Numerade Educator

Problem 25

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that if $h>-1$ , then $1+n h \leq(1+h)^{n}$ for all non-negative integers $n .$ This is called Bernoulli's inequality.

Prathan J.
Numerade Educator

Problem 26

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Suppose that $a$ and $b$ are real numbers with $0< b <a$ . Prove that if $n$ is a positive integer, then $a^{n}-b^{n} \leq$ $n a^{n-1}(a-b)$

Prathan J.
Numerade Educator

Problem 27

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that for every positive integer $n,$
$\quad 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1)$

Prathan J.
Numerade Educator

Problem 28

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Prove that $n^{2}-7 n+12$ is nonnegative whenever $n$ is an integer with $n \geq 3$ .

Prathan J.
Numerade Educator

Problem 29

In Exercises 29 and $30, H_{n}$ denotes the $n$ th harmonic number.
Prove that $H_{2^{n}} \leq 1+n$ whenever $n$ is a nonnegative integer.

Prathan J.
Numerade Educator

Problem 30

In Exercises 29 and $30, H_{n}$ denotes the $n$ th harmonic number.
Prove that
$$
H_{1}+H_{2}+\cdots+H_{n}=(n+1) H_{n}-n
$$

Prathan J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

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 J.
Numerade Educator

Problem 38

Use mathematical induction in Exercises $38-46$ to prove results about sets.
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 J.
Numerade Educator

Problem 39

Use mathematical induction in Exercises $38-46$ to prove results about sets.
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 J.
Numerade Educator

Problem 40

Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B$ are sets, then
$$
\begin{aligned}\left(A_{1} \cap A_{2} \cap \cdots \cap A_{n}\right) \cup B & \\=\left(A_{1} \cup B\right) \cap\left(A_{2} \cup B\right) \cap \cdots \cap\left(A_{n} \cup B\right) \end{aligned}
$$

Prathan J.
Numerade Educator

Problem 41

Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B$ are sets, then
$$
\begin{aligned}\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right) \cap B & \\=\left(A_{1} \cap B\right) \cup\left(A_{2} \cap B\right) \cup \cdots \cup\left(A_{n} \cap B\right) \end{aligned}
$$

Prathan J.
Numerade Educator

Problem 42

Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B$ are sets, then
$$
\begin{aligned}\left(A_{1}-B\right) \cap\left(A_{2}-B\right) \cap \cdots \cap\left(A_{n}-B\right) \\=\left(A_{1} \cap A_{2} \cap \cdots \cap A_{n}\right)-B \end{aligned}
$$

Prathan J.
Numerade Educator

Problem 43

Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ are subsets of a universal set $U,$ then
$$
\bigcup_{k=1}^{n} A_{k}=\bigcap_{k=1}^{n} \overline{A_{k}}
$$

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Problem 44

Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B$ are sets, then
$$
\begin{aligned}\left(A_{1}-B\right) \cup\left(A_{2}-B\right) \cup \cdots \cup\left(A_{n}-B\right) \\=\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right)-B \end{aligned}
$$

Prathan J.
Numerade Educator

Problem 45

Use mathematical induction in Exercises $38-46$ to prove results about sets.
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 $2 .$

Prathan J.
Numerade Educator

Problem 46

Use mathematical induction in Exercises $38-46$ to prove results about sets.
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 $3 .$

Prathan J.
Numerade Educator

Problem 47

In Exercises 47 and 48 we consider the problem of placing towers along a straight road, so that every building on the road receives cellular service. Assume that a building receives cellular service if it is within one mile of a tower.
Devise a greedy algorithm that uses the minimum number of towers possible to provide cell service to $d$ buildings located at positions $x_{1}, x_{2}, \ldots, x_{d}$ from the start of the road. [Hint: At each step, go as far as possible along the road before adding a tower so as not to leave any buildings without coverage.]

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Problem 48

In Exercises 47 and 48 we consider the problem of placing towers along a straight road, so that every building on the road receives cellular service. Assume that a building receives cellular service if it is within one mile of a tower.
Use mathematical induction to prove that the algorithm you devised in Exercise 47 produces an optimal solution, that is, that it uses the fewest towers possible to provide cellular service to all buildings.

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Problem 49

Exercises $49-51$ present incorrect proofs using mathematical induction. You will need to identify an error in reasoning in each exercise.
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.

Prathan J.
Numerade Educator

Problem 50

Exercises $49-51$ present incorrect proofs using mathematical induction. You will need to identify an error in reasoning in each exercise.
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, we have $\sum_{i=1}^{n+1} i=\left(n+\frac{1}{2}\right)^{2} / 2+n+1=$ $\left(n^{2}+n+\frac{1}{4}\right) / 2+n+1=\left(n^{2}+3 n+\frac{9}{4}\right) / 2=$ $\left(n+\frac{3}{2}\right)^{2} / 2=\left[(n+1)+\frac{1}{2}\right]^{2} / 2,$ completing the inductive step.

Prathan J.
Numerade Educator

Problem 51

Exercises $49-51$ present incorrect proofs using mathematical induction. You will need to identify an error in reasoning in each exercise.
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 J.
Numerade Educator

Problem 52

Suppose that $m$ and $n$ are positive integers with $m>n$ and $f$ is a function from $\{1,2, \ldots, m\}$ to $\{1,2, \ldots, n\} .$ Use mathematical induction on the variable $n$ to show that $f$ is not one-to-one.

Prathan J.
Numerade Educator

Problem 53

Use mathematical induction to show that $n$ people can divide a cake (where each person gets one or more separate pieces of the cake) so that the cake is divided fairly, that is, in the sense that each person thinks he or she got at least $(1 / n)$ th of the cake. [Hint: For the inductive step, take a fair division of the cake among the first $k$ people, have each person divide their share into what this person thinks are $k+1$ equal portions, and then have the $(k+1)$ st person select a portion from each of the $k$ people. When showing this produces a fair division for $k+1$ people, suppose that person $k+1$ thinks that person $i$ got $p_{i}$ of the cake, where $\sum_{i=1}^{k} p_{i}=1 . ]$

Prathan J.
Numerade Educator

Problem 54

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 J.
Numerade Educator

Problem 55

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 that denote the row number and the column number of the square, respectively. Use mathematical induction to show that a knight starting at $(0,0)$ can visit every square using a finite sequence of moves. [Hint: Use induction on the variable $s=m+n . ]$

Prathan J.
Numerade Educator

Problem 56

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 J.
Numerade Educator

Problem 57

(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 J.
Numerade Educator

Problem 58

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

Prathan J.
Numerade Educator

Problem 59

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 J.
Numerade Educator

Problem 60

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 J.
Numerade Educator

Problem 61

Show that
$\left[\left(p_{1} \rightarrow p_{2}\right) \wedge\left(p_{2} \rightarrow p_{3}\right) \wedge \cdots \wedge\left(p_{n-1} \rightarrow p_{n}\right)\right]$
$\quad \rightarrow\left[\left(p_{1} \wedge p_{2} \wedge \cdots \wedge p_{n-1}\right) \rightarrow p_{n}\right]$
is a tautology whenever $p_{1}, p_{2}, \ldots, p_{n}$ are propositions, where $n \geq 2$

Prathan J.
Numerade Educator

Problem 62

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

Prathan J.
Numerade Educator

Problem 63

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 $A \geq G$ .

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Problem 64

Use mathematical induction to prove Lemma 3 of Section $4.3,$ which states that if $p$ is a prime and $p | a_{1} a_{2} \cdots a_{n},$ where $a_{i}$ is an integer for $i=1,2,3, \ldots, n,$ then $p | a_{i}$ for some integer $i .$

Prathan J.
Numerade Educator

Problem 65

Show that if $n$ is a positive integer, then
$$
_{\left\{a_{1}, \ldots, a_{k}\right\} \subseteq\{1,2, \ldots, n\}} \frac{1}{a_{1} a_{2} \cdots a_{k}}=n
$$
(Here the sum is over all nonempty subsets of the set of the $n$ smallest positive integers.)

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Problem 66

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 J.
Numerade Educator

Problem 67

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 $A_{j}$ or $A_{j}$ is a subset of $A_{i},$ 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 J.
Numerade Educator

Problem 68

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 question asking 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. [Hint: First ask a question to eliminate one person as a celebrity. Then use the inductive hypothesis to identify a potential celebrity. Finally, ask two more questions to determine whether that person is actually a celebrity. $]$

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Problem 69

Suppose there are $n$ people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for $G(n),$ the minimum number of telephone calls that are needed for all $n$ people to learn about all the scandals. Exercises $69-71$ deal with the gossip problem.
Find $G(1), G(2), G(3),$ and $G(4)$

Prathan J.
Numerade Educator

Problem 70

Suppose there are $n$ people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for $G(n),$ the minimum number of telephone calls that are needed for all $n$ people to learn about all the scandals. Exercises $69-71$ deal with the gossip problem.
Use mathematical induction to prove that $G(n) \leq 2 n-4$ for $n \geq 4 .$ [Hint: In the inductive step, have a new person call a particular person at the start and at the end. $]$

Prathan J.
Numerade Educator

Problem 71

Suppose there are $n$ people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for $G(n),$ the minimum number of telephone calls that are needed for all $n$ people to learn about all the scandals. Exercises $69-71$ deal with the gossip problem.
Prove that $G(n)=2 n-4$ for $n \geq 4$

Prathan J.
Numerade Educator

Problem 72

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.]

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Problem 73

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 J.
Numerade Educator

Problem 74

Sometimes we cannot use mathematical induction to prove a result we believe to be true, but we can use mathematical induction to prove a stronger result. Because the inductive hypothesis of the stronger result provides more to work with, this process is called inductive loading. We use inductive loading in Exercise $74-76$ .
Show that we cannot use mathematical induction to prove that $\sum_{j=1}^{n} 1 / j^{2}<2$ for all positive integers $n$ , but that this inequality is a consequence of the inequality proved by mathematical induction in Exercise $19 .$

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Problem 75

Sometimes we cannot use mathematical induction to prove a result we believe to be true, but we can use mathematical induction to prove a stronger result. Because the inductive hypothesis of the stronger result provides more to work with, this process is called inductive loading. We use inductive loading in Exercise $74-76$ .
Suppose that we want to prove that
$$
\sum_{j=1}^{n} j /(j+1) !<1
$$
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
$$
\sum_{j=1}^{n} j /(j+1) ! \leq 1-1 /(n+1) !
$$
for all positive integers $n,$ implying that the weaker inequality is also true.

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Problem 76

Sometimes we cannot use mathematical induction to prove a result we believe to be true, but we can use mathematical induction to prove a stronger result. Because the inductive hypothesis of the stronger result provides more to work with, this process is called inductive loading. We use inductive loading in Exercise $74-76$ .
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
$$
\frac{1}{2} \cdot \frac{3}{4} \dots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n+1}}
$$
for all integers $n$ 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.

Prathan J.
Numerade Educator

Problem 77

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

Prathan J.
Numerade Educator

Problem 78

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

Prathan J.
Numerade Educator

Problem 79

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

Prathan J.
Numerade Educator

Problem 80

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

Prathan J.
Numerade Educator

Problem 81

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 82

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 \not{|} $ $n.$

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Problem 83

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

Prathan J.
Numerade Educator

Problem 84

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 J.
Numerade Educator

Problem 85

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 integers $k$ with $k \geq b$ .

Prathan J.
Numerade Educator