수학적 귀납법

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