수열의 귀납적 정의

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 $n \in \mathbb{N} \cup \left\{ 0 \right\}$, $a_n = n!$.

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

For each $n \in \mathbb{N}$, $f_{4n}$ is a multiple of 3.

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

For each $n \in \mathbb{N}$, $f_{5n}$ is a multiple of 5.

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

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

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

For each $n \in \mathbb{N}$, $f_1 + f_3 + \cdots + f_{2n - 1} = f_{2n}$.

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

For each $n \in \mathbb{N}$, $f_2 + f_4 + \cdots + f_{2n} = f_{2n + 1} - 1$.

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

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

The quadratic formula can be used to show that $\alpha = \dfrac{1 + \sqrt{5}}{2}$ and $\beta = \dfrac{1 - \sqrt{5}}{2}$ are the two real number solutions of the quadratic equation $x^2 - x - 1 = 0$. Notice that this implies that

$$\begin{aligned}\alpha^2 & = \alpha + 1, \text{ and } \\ \beta^2 & = \beta + 1\end{aligned}$$

. It may be surprising to find out that these two irrational numbers are closely related to the Fibonacci numbers.

Verify that $f_1 = \dfrac{\alpha^1 - \beta^1}{\alpha - \beta}$ and that $f_2 = \dfrac{\alpha^2 - \beta^2}{\alpha - \beta}$.

The quadratic formula can be used to show that $\alpha = \dfrac{1 + \sqrt{5}}{2}$ and $\beta = \dfrac{1 - \sqrt{5}}{2}$ are the two real number solutions of the quadratic equation $x^2 - x - 1 = 0$. Notice that this implies that

$$\begin{aligned}\alpha^2 & = \alpha + 1, \text{ and } \\ \beta^2 & = \beta + 1\end{aligned}$$

. It may be surprising to find out that these two irrational numbers are closely related to the Fibonacci numbers.

Use induction to prove that for each natural number $n$, if $\alpha = \dfrac{1 + \sqrt{5}}{2}$ and $\beta = \dfrac{1 - \sqrt{5}}{2}$, then $f_n = \dfrac{\alpha^n - \beta^n}{\alpha - \beta}$.

This formula for the $n^{th}$ Fibonacci number is known as Binet's formula,

named after the French mathematician Jacques Binet (1786 — 1856).

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$ , assume that $a_1 = 1$, $a_2 = 5$, and that for each $n \in \mathbb{N}$ with $n\geq 2$, $a_{n + 1} = a_n + 2a_{n-1}$. Prove that for each natural number $n$, $a_n = 2^n + (-1)^n$.

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$ , assume that $a_1 = 1$ and that for each $n \in \mathbb{N}$, $a_{n + 1} = \sqrt {5 + a_n }$.

Prove that for each $n \in \mathbb{N}$, $a_n \lt 3$.

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$ , assume that $a_1 = 1$, $a_2 = 1$, and that for each $n \in \mathbb{N}$, $a_{n + 2} = \dfrac{1}{2}\left( {a_{n + 1} + \dfrac{2}{{a_n }}} \right)$.

Prove that for each $n \in \mathbb{N}$, $1 \leq a_n \leq 2$.

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$, assume that $a_1=1$, $a_2=1$, $a_3=1$, and for that each natural number $n$,

$$a_{n+3} = a_{n+2} + a_{n+1} + a_n$$

.

Prove that for each natural number $n$ with $n>1$, $a_n \leq 2^{n-2}$.

For the sequence $a_1, a_2, \ldots, a_n, \ldots$ , assume that $a_1 = 1$, $a_2 = 1$, and for each $n \in \mathbb{N}$, $a_{n+2} = a_{n+1} + 3a_n$. Determine which terms in this sequence are divisible by 4 and prove that your answer is correct.

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 $n$, $L_n = 2f_{n+1} - f_n$.

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 $n \in \mathbb{N}$ with $n \geq 2$, $5f_n = L_{n - 1} + L_{n + 1}$.

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 $n \in \mathbb{N}$ with $n \geq 3$, $L_n = f_{n + 2} - f_{n - 2}$.