수학적 귀납법 연습문제 — Sundstrom, Mathematical Reasoning · 4.1절 · 8a번

Sundstrom, Mathematical Reasoning · 4.1절 · 8a번영어 원문

Use mathematical induction to prove each of the following:

For each natural number $n$ , 3 divides $\left( {4^n - 1} \right)$ .

정답·풀이 보기 (영어 원문)

The key to the inductive step is that if $4^k = 1 + 3m$ , then $4^k \cdot 4 = 4 ( 1 + 3m )$ , which implies that

$4^{k+1} - 1 = 3 (1 + 4m )$

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 $2^n$ greater than $\left(n + 1 \right)^2$ ?

다 풀었으면 정답 확인

For each natural number $n$ , if $n \geq 6$ , then $2^n > (n + 1 )^2$ .

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