수열 전체 문제

Can each natural number greater than or equal to 4 be written as the sum of at least two natural numbers, each of which is a 2 or a 3? Justify your conclusion. For example, $7 = 2 + 2 + 3$, and $17 = 2 + 2 + 2 + 2 + 3 + 3 + 3$.

Can each natural number greater than or equal to 6 be written as the sum of at least two natural numbers, each of which is a 2 or a 5? Justify your conclusion. For example, $6 = 2 + 2 + 2$, $9 = 2 + 2 + 5$, and $17 = 2 + 5 + 5 + 5$.

Use mathematical induction to prove the following proposition:

Let

x

be a real number with

x > 0

. Then for each natural number

n

with

n \geq 2

,

\left( {1 + x} \right)^n > 1 + nx

.

Explain where the assumption that $x > 0$ was used in the proof.

Prove that for each odd natural number $n$ with $n \geq 3$,

$$\left( {1 + \frac{1}{2}} \right) \left( {1 - \frac{1}{3}} \right) \left( {1 + \frac{1}{4}} \right) \cdots \left( {1 + \frac{{\left( { - 1} \right)^n }}{n}} \right) = 1$$

.

Prove that for each natural number $n$, any set with $n$ elements has $\dfrac{n \left( {n-1} \right)}{2}$ two-element subsets.

Prove or disprove each of the following propositions:

For each $n \in \mathbb{N}$, $\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \cdots + \dfrac{1}{n (n + 1 )} = \dfrac{n}{n+1}$.

Prove or disprove each of the following propositions:

For each natural number $n$ with $n \geq 3$,

$$\frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \cdots + \frac{1}{n (n + 1 )} = \frac{n-2}{3n + 3}$$

.

Prove or disprove each of the following propositions:

For each $n \in \mathbb{N}$, $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + n (n + 1 ) = \dfrac{n (n + 1 ) (n + 2 )}{3}$.

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 $n \in \mathbb{N}$, $1 \leq a_n \leq 2$.