# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 +: $$m+0=m\quad \text{for all}\quad m \in \mathbb{N} \\ m+(k+1) = (m+k)+1$$ in ...
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### $3^a\mid s(n) \Rightarrow 3^a\mid n$

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 ...
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### 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 ...
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### 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 $n ∈ \mathbb{N}, a_n = 2^n$
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### 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 + ...
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### 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 ...
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### 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 $100$.
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### 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 ...
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### 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 ...
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### 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 ...
### 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: $a+0=a$ ...