수학적 귀납법 연습문제 — Sundstrom, Mathematical Reasoning · 4.2절 · 8번

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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