수열의 귀납적 정의 연습문제 — Sundstrom, Mathematical Reasoning · 4.3절 · 2f번

Sundstrom, Mathematical Reasoning · 4.3절 · 2f번영어 원문

Assume that $f_1 ,f_2 , \ldots ,f_n , \ldots$ are the Fibonacci numbers. Prove each of the following:

For each $n \in \mathbb{N}$ , $f_1^2 + f_2^2 + \cdots + f_n^2 = f_n f_{n + 1}$ .

정답·풀이 보기 (영어 원문)

Let $P( n )$ be, “ $f_1^2 + f_2^2 + \cdots + f_n^2 = f_n f_{n + 1}$ .” For the basis step, we notice that $f_1^2 = 1$ and $f_1 \cdot f_2 = 1$ and hence, $P(1)$ is true. For the inductive step, we need to prove that if $P(k)$ is true, then $P(k+1)$ is true. That is, we need to prove that if $f_1^2 + f_2^2 + \cdots + f_k^2 = f_k f_{k + 1}$ , then $f_1^2 + f_2^2 + \cdots + f_k^2 + f_{k+1}^2 = f_{k+1} f_{k + 2}$ . To do this, we can use

$\begin{aligned}\left( f_1^2 + f_2^2 + \cdots + f_k^2 \right) + f_{k+1}^2 & = f_k f_{k + 1} + f_{k+1}^2 \\ f_1^2 + f_2^2 + \cdots + f_k^2 + f_{k+1}^2 & = f_{k+1} \left( f_k + f_{k+1} \right) \\ & = f_{k+1} f_{k+2}\end{aligned}$

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.3절 · 1번영어 원문

For the sequence $a_{0}, a_1 ,a_2 , \ldots , a_n , \ldots$ , assume that $a_0 = 1$ and that for each $n \in \mathbb{N} \cup \left\{ 0 \right\}$ , $a_{n + 1} = \left( {n + 1} \right)a_n$ . Use mathematical induction to prove that for each $n \in \mathbb{N} \cup \left\{ 0 \right\}$ , $a_n = n!$ .

다 풀었으면 정답 확인

Let $P(n)$ be $a_n = n!$ . Since $a_0 = 1$ and $0 ! = 1$ , we see that $P(0)$ is true. For the inductive step, we assume that $k \in \mathbb{N} \cup \{0\}$ and that $P(k)$ is true or that $a_k = k!$ .

$\begin{aligned}a_{k+1} & = \left( k + 1 \right)a_k \\ & = \left( k + 1 \right) k! \\ & = \left( k + 1 \right)!\end{aligned}$

. This proves the inductive step that if $P(k)$ is true, then $P(k+1)$ is true.

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