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수학적 귀납법

Name: 대수 17강 수학적 귀납법
Uploaded: 2026-10-27T23:47:50.229Z
Duration: 4 min 43 s
Description: 자연수에 대한 명제를 증명하는 두 단계

자연수에 대한 명제를 증명하는 두 단계

4:43 · 한국어 자막 있음(재생기 메뉴) · 연습문제 원문: Ted Sundstrom, Mathematical Reasoning: Writing and Proof, Version 3 (PreTeXt edition by Ian Curtis). Grand Valley State University Libraries. https://scholarworks.gvsu.edu/books/28 — 4.1절 연습문제 3(a) (https://gvsuoer.github.io/sundstrom-textbook/S_mathinduction.html#exer_sec51-3) / 4.2절 연습문제 1(a) (https://gvsuoer.github.io/sundstrom-textbook/S_otherinduction.html#exer_sec52-1). 문제를 우리말로 옮겼으며, 원문은 CC BY-NC-SA 3.0 Unported, 이 영상은 같은 조건의 이후 판인 CC BY-NC-SA 4.0 으로 공개합니다.

핵심 정리

자연수 $n$ 에 대한 명제 $p(n)$ 이 모든 자연수 $n$ 에 대하여 참임을 보이려면 (ⅰ) $n=1$ 일 때 참이고 (ⅱ) $n=k$ 일 때 참이라고 가정하면 $n=k+1$ 일 때도 참임을 보이면 됩니다.

예: $1+2+\cdots+n=\frac{n(n+1)}{2}$ — $n=k$ 일 때 성립한다고 가정하고 양변에 $k+1$ 을 더하면 $\frac{k(k+1)}{2}+(k+1)=\frac{(k+1)(k+2)}{2}$ 이므로 $n=k+1$ 일 때도 성립합니다.

$n\ge m$ 인 자연수에 대한 명제는 (ⅰ)을 $n=m$ 에서 확인합니다. 부등식은 (ⅱ)에서 가정한 부등식의 양변에 같은 양을 더하거나 같은 양수를 곱해 비교합니다.

연습문제

원문 · Sundstrom, Mathematical Reasoning: Writing and Proof (CC BY-NC-SA 3.0)

1.Sundstrom, Mathematical Reasoning · 4.1절 · 2b번영어 원문

Explain the following.

Is the empty set inductive?

정답·풀이 보기 (영어 원문)

The empty set is inductive (why?).

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.1절 · 2a번영어 원문

Explain the following.

Can a finite, nonempty set be inductive?

다 풀었으면 정답 확인

A finite nonempty set is not inductive (why?).

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# 수학적 귀납법계산자동 분류

2.Sundstrom, Mathematical Reasoning · 4.1절 · 14번영어 원문

Use mathematical induction to prove that the sum of the cubes of any three consecutive natural numbers is a multiple of 9.

정답·풀이 보기 (영어 원문)

Three consecutive natural numbers may be represent by $n$ , $n + 1$ , and $n + 2$ , where $n$ is a natural number. For the inductive step, think before you try to do a lot of algebra. You should be able to complete a proof of the inductive step by expanding the cube of only one expression.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.2절영어 원문

Consider the following question:

For which natural numbers

n

do there exist nonnegative integers

x

and

y

such that

n = 3x + 5y

?

To help answer this question, we will let $\mathbb{Z}^* = \left\{ {x \in \mathbb{Z} \mid x \geq 0} \right\}$ , and let $P \left( n \right)$ be There exist $x,y \in \mathbb{Z}^*$ such that $n = 3x + 5y$ . Notice that $P(1)$ is false since if both $x$ and $y$ are zero, then $3x + 5y = 0$ and if either $x > 0$ or $y > 0$ , then $3x + 5y \geq 3$ . Also notice that $P(6)$ is true since $6 = 3 \cdot 2 + 5 \cdot 0$ and $P(8)$ is true since $8 = 3 \cdot 1 + 5 \cdot 1$ .

Explain why $P(9)$ , $P(10)$ , $P(11)$ , and $P(12)$ are true.

다 풀었으면 정답 확인

Construct the following table and use it to answer the first two questions. The table shows that $P\left( 3 \right)$ , $P\left( 5 \right)$ , and $P\left( 6 \right)$ are true. We can also see that $P(2)$ , $P(4)$ , and $P(7)$ are false. It also appears that if $n \in \mathbb{N}$ and $n \geq 8$ , then $P\left( n \right)$ is true.

| $x$ | 0 | 1 | 2 | 3 | 4 | 0 | 1 | 2 | 0 | 1 | 1 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | $y$ | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 3 | 3 | | $3x + 5y$ | 0 | 3 | 6 | 9 | 12 | 5 | 8 | 11 | 10 | 13 | 18 |

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# 수학적 귀납법증명·논술자동 분류

3.Sundstrom, Mathematical Reasoning · 4.2절영어 원문

Formulate a conjecture (with an appropriate quantifier) that can be used as an answer to each of the following questions.

For which natural numbers $n$ is $\left( 1 + \dfrac{1}{n} \right)^n$ greater than 2.5?

정답·풀이 보기 (영어 원문)

For each natural number $n$ , if $n \geq 6$ , then $\left( 1 + \dfrac{1}{n} \right)^n > 2.5$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.2절영어 원문

Formulate a conjecture (with an appropriate quantifier) that can be used as an answer to each of the following questions.

For which natural numbers $n$ is $3^n$ greater than $1 + 2^n$ ?

다 풀었으면 정답 확인

For each natural number $n$ , if $n \geq 2$ , then $3^n > 1 + 2^n$ .

이 문제로 이어서 풀기

# 수학적 귀납법계산자동 분류

4.Sundstrom, Mathematical Reasoning · 4.2절 · 8번영어 원문

Can each natural number greater than or equal to 4 be written as the sum of at least two natural numbers, each of which is a 2 or a 3? Justify your conclusion. For example, $7 = 2 + 2 + 3$ , and $17 = 2 + 2 + 2 + 2 + 3 + 3 + 3$ .

정답·풀이 보기 (영어 원문)

Let $P( n )$ be, “The natural number $n$ can be written as a sum of natural numbers, each of which is a 2 or a 3.” Verify that $P( 4 ), P( 5 ), P( 6 )$ , and $P( 7 )$ are true. To use the Second Principle of Mathematical Induction, assume that $k \in \mathbb{N}, k \geq 5$ and that $P( 4 ), P( 5 ), \ldots , P( k )$ are true. Then notice that

$k + 1 = ( {k - 1} ) + 2$

. Since $k - 1 \geq 4$ , we have assumed that $P( {k - 1} )$ is true. This means that $(k-1)$ can be written as a sum of natural numbers, each of which is a 2 or a 3. Since $k + 1 = (k - 1) + 2$ , we can conclude that $(k+1)$ can be written as a sum of natural numbers, each of which is a 2 or a 3. This completes the proof of the inductive step.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.2절 · 5번영어 원문

Is the following proposition true or false? Justify your conclusion.

For each nonnegative integer

8^n \mid \left( {4n} \right)!

다 풀었으면 정답 확인

Let $P( n )$ be the predicate, “ $8^n \mid ( 4n )!$ .” Verify that $P( 0 ), P( 1 ), P( 2 )$ , and $P( 3 )$ are true. For the inductive step, the following fact about factorials may be useful:

$\begin{aligned}[4 (k + 1) ]! & = ( 4k + 4 )! \\ & = (4k + 4)(4k + 3)(4k + 2)(4k + 1) ( 4k )!\end{aligned}$

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# 수학적 귀납법증명·논술자동 분류

「수학적 귀납법」 문제 더 풀기 (42문제)

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