수학적 귀납법 연습문제 — Sundstrom, Mathematical Reasoning · 4.1절 · 14번

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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