3
votes
5answers
568 views

The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each ...
1
vote
2answers
53 views

proof by induction that every non-zero natural number has a predecessor

I am trying to prove by induction that every non-zero natural number has at least one predecessor. However, I don't know what to use as a base case, since 0 is not non-zero and I haven't yet ...
0
votes
2answers
79 views

Prove the commutativity property of addition of natural numbers by induction

the background I'm allowed to deal with to solve this problem is as follows: Definition of +: \begin{equation} m+0=m\quad \text{for all}\quad m \in \mathbb{N} \\ m+(k+1) = (m+k)+1 \end{equation} in ...
1
vote
2answers
30 views

$3^a\mid s(n) \Rightarrow 3^a\mid n$

This is not a homework question, neither a championship problem (as far as I've searched in the net), and it came up noticing a singular pattern, involving the powers of $3$: "Prove or disprove that ...
6
votes
4answers
82 views

Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research ...
0
votes
2answers
76 views

Prove $a_n = 2^n$ by strong induction [closed]

Given the sequence $a_n = a_{n-1} + ... + a_0 + 1$, prove by strong induction that for any $n ∈ \mathbb{N}, a_n = 2^n$
1
vote
1answer
27 views

Proof $\prod_{i = 1}^n \frac{n + i}{2i-3} = 2^n(1-2n)$ using inducction

i'm trying to solve this, using induction. The base step is easy, there's no difficult there. The problem comes in the inductive step, I got to demonstrate that: $$\prod_{i = 1}^{n+1} \frac{n+ 1 + ...
1
vote
1answer
65 views

Properties of Natural Numbers and Mathematical Induction

When working with natural numbers how to check that the property we consider is "permissible" to speak about? And not like the property "The smallest positive integer not definable in under eleven ...
0
votes
1answer
42 views

How Can I find the summation of divisors of $n^p$.

For Example $n=8$ and $p=2$. So $n^p=64$. And the summation of divisors is $1+2+4+8+16+32+64=127$. But the problem arises when $n=10^6$ and $p=10^6$. Remember u can modulus the result by $100$.
1
vote
1answer
71 views

Prove any $k\in\mathbb{N}$ can be created using sum of $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$

I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition. My attempt was just bad... I've ...
5
votes
1answer
2k views

Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$

I'm working on a task where I'm a bit unsure if the answer I've got is correct. Here is the task: Show by induction that the following assertion is true for all natural numbers $n$ $n^3 - ...
1
vote
1answer
66 views

Induction on natural numbers

My textbook, Logic and Discrete Mathematics by Grassman and Tremblay, has an example which I can't wrap my head around (example 3.4; page 127). It shows that for all $n$, $2(n+2)\le(n+2)^2$. As the ...
0
votes
1answer
86 views

Induction, how often?

I have the following definition: $\quad p(x) \iff (x=0 \vee$ $\quad\quad\quad\quad\quad\quad\exists y\ (x=y+2\ \&\ p(y)) \vee$ $\quad\quad\quad\quad\quad\quad(x=1\ \&\ \forall y\ p(y*2)))$ ...
4
votes
4answers
183 views

Prove $3^n \ge n^3$ by induction

Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$. I can do this myself, but can't figure out any kind of "beautiful" way to do it. The way I do it is: Assume $3^n \ge n^3$ Now, $(n+1)^3 = n^3 + ...
1
vote
4answers
336 views

How can I expand mathematical induction to rational numbers?

I know mathematical induction can be used to prove that a statements is true for all natural numbers (or those belonging to a certain subset of N). However, it is pretty obvious, unless I'm terribly ...
1
vote
1answer
69 views

Prove inequality by induction

Once again, I'm stuck in a demonstration by induction, this time, it's really proving that an inequality is valid. So, here is the inequality: Prove that $\binom{2n}{n} \geq (n+5)^2 \ \forall n ...
1
vote
2answers
61 views

Is this inequality property true?

I'm having some trouble defining weather this inequality is true or not... Basically, I wanted to know if its true that if $a \geq b$ and $c \geq d \Rightarrow a + c \geq b + d$ Well, basically ...
2
votes
1answer
94 views

associativity of multipication of natural numbers

I am trying to prove by induction the associativity of natural numbers. It is easy to see that if $n,m\in \mathbb{N}$, then $(mn)1=m(n1)$. If $p\in \mathbb{N}$ is such that $(mn)p= m(np)$, then ...
2
votes
2answers
90 views

How to prove that $+$ is commutative on the natural numbers?

Let $N$ be a non empty set. Let $s:N\to N$ a function satisfying: there is only one element in $N-s(N)$ (denoted by $1$); $s$ is injective; for any subset $X\subset N$, if $1\in X$ and $(n\in N ...
8
votes
3answers
1k views

Consistency of Peano axioms (Hilbert's second problem)?

(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary ...
8
votes
3answers
366 views

Commutativity of multiplication in $\mathbb{N}$

I'm trying to prove that $a\cdot b=b\cdot a$ when $a$ and $b$ are two natural numbers. In the rest of this question I'm using $a'$ for the successor of $a$. Addition is defined as: $a+0=a$ ...
1
vote
2answers
218 views

Strict ordering on natural numbers

I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way: They prove ...