수열의 귀납적 정의 연습문제 — Sundstrom, Mathematical Reasoning · 4.3절 · 13a번

Sundstrom, Mathematical Reasoning · 4.3절 · 13a번영어 원문

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$ , assume that $a_1 = 1$ , $a_2 = 3$ , and that for each $n \in \mathbb{N}$ , $a_{n + 2} = 3a_{n + 1} - 2a_n$ .

Calculate $a_3$ through $a_6$ .

정답·풀이 보기 (영어 원문)

$a_3 = 7$ , $a_4 =15$ , $a_5 =31$ , $a_6 =63$

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.3절 · 18a번영어 원문

The Lucas numbers

are a sequence of natural numbers $L_1, L_2, L_3, \ldots, L_n, \ldots$ , which are defined recursively as follows:

$L_1 = 1$ and $L_2 = 3$ , and - For each natural number $n$ , $L_{n+2} = L_{n+1} + L_n$ .

List the first 10 Lucas numbers and the first ten Fibonacci numbers and then prove each of the following propositions. The Second Principle of Mathematical Induction may be needed to prove some of these propositions.

For each natural number $n$ , $L_n = 2f_{n+1} - f_n$ .

다 풀었으면 정답 확인

Let $P(n)$ be, “ $L_n = 2f_{n+1} - f_n$ .” First, verify that $P(1)$ and $P(2)$ are true. Now let $k$ be a natural number with $k \geq 2$ and assume that $P(1)$ , $P(2)$ , …, $P(k)$ are all true. Since $P(k)$ and $P(k-1)$ are both assumed to be true, we can use them to help prove that $P(k+1)$ must then be true as follows:

$\begin{aligned}L_{k+1} & = L_k + L_{k-1} \\ & = \left( 2f_{k+1} - f_k \right) + \left( 2f_k - f_{k-1} \right) \\ & = 2 \left( f_{k+1} + f_k \right) - \left( f_k + f_{k-1} \right) \\ & = 2f_{k+2} - f_{k+1}\end{aligned}$

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