수학적 귀납법 연습문제 — Sundstrom, Mathematical Reasoning · 4.1절 · 2a번

Sundstrom, Mathematical Reasoning · 4.1절 · 2a번영어 원문

Explain the following.

Can a finite, nonempty set be inductive?

정답·풀이 보기 (영어 원문)

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

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.1절 · 8a번영어 원문

Use mathematical induction to prove each of the following:

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

다 풀었으면 정답 확인

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

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