Induction in general is the inference from the particular to the general. Mathematical induction is not true induction, but is a form of deductive reasoning. Its most common use is induction over well ordered sets, such as natural numbers, or ordinals. While induction can be expanded to class ...

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My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
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135 views

Cauchy induction: are there examples of cases where choosing an integer other than $2$ is a better strategy?

Cauchy induction, sometimes called backwards induction, works as follows: show that $p(1)$ is true show that $p(n)$ implies $p(2n)$ (which inductively implies $p(2^n)$ is true) show that $p(n)$ ...
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49 views

Prove $\forall n\in\mathbb{N}, \exists m\in\mathbb{N}; n=\pm1^2\pm2^2\pm\cdots\pm m^2.$

And we choose the positive and negative signs in a way that the equation becomes true. I think it can be proved with mathematical induction. So here's how I begin: For $n=1$, $1=+1^2$ which is true. ...
4
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160 views

Efficiently count possible nim-like moves

Consider $n$ piles of coins, with pile $i$ having $a_i$ coins. A valid move is to remove zero or more coins from each of the piles, with the constraint that atleast one pile should remain unchanged, ...
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111 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any ...
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39 views

Combinatorics, equality, $n$-permutations with $k$ cycles

Let $b_r(n,k)$ denote the number of $n$-permuations with $k$ cycles in which all numbers: $1,...,r$ are in one cycle. Prove that for $n \ge r$, $\sum _{k=1} ^n b_r(n,k) x^k = (r-1)! ...
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67 views

Is this a valid statement that would imply the Collatz Conjecture?

Let $f$ denote the Collatz transformation: $f(x) = \left\{ \begin{array}{ll} {x\over 2} & \quad x\equiv 0 \mod 2 \\ 3x+1 & \quad x \equiv 1\mod 2 ...
3
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81 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
3
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71 views

Induction on Primitive Recursive Function

The set $F_{n}$ of primitive recursive function symbols of arity $n$ can be defined inductively as: \begin{align} & Z, \text{Succ} \in F_{1} & \\ &\pi_{j}^{n} \in F_{n} \quad \text{for ...
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471 views

Induction in proof of multiplicativity of Euler totient function

(Updated below) I'm working through John Stillwell's Elements of Algebra, and while his exercises are generally crafted to be not too difficult, there's one that I don't even understand what it's ...
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63 views

Prove, using the method of mathematical induction that the following holds true

For natural numbers $n\ge1$ show the following inequality using induction. $$n^{1/n}\le 1+\sqrt{\frac{2}{n}}$$
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25 views

Confusion on particular step in this weak induction proof

I am studying for a midterm, and I came across this proof. Use mathematical induction(weak) to show that for all integers n $\geq$ 2 , if x$_{1}$, x$_{2}$, ... x$_{n}$ are strictly between 0 and 1 ...
2
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48 views

IS this proof by induction of the hand shake lemma correct?

Proof by induction that the sum of degrees of vertexes in an undirected graph equals two times the number edges, where $V$ is the set of vertexes and $E$ is an edge multiset: $$\sum_{v ∈ V} deg(v) = ...
2
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80 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
2
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53 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$$ which can be proved combinatorically whether one particular element(among the $n$) is ...
2
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146 views

Homework Question for a 15 year old

My younger brother(age: 14 years 7 months) and his classmates were given a set of eight questions by his class-teacher, which included the following two questions: (i) Find, if you can, the fallacy ...
2
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57 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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79 views

Simple knapsack with arbitrary weights: Algorithm won't work, but my proof by induction doesn't agree.

We want to solve the simple knapsack problem: We're given a set of $n$ positive item weights, which are unique integers $\{w_1, \ldots , w_n\}$, and an integer $C > 0$, representing the capacity of ...
2
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0answers
81 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
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96 views

Rearranging numbered cards to reverse their order

I have been thinking about this question for a long time, but I can't solve it. Here is the question: We have $9$ cards, with numbers one to nine written on them (in the order $1, 2, \ldots , 9$). ...
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42 views

How to proceed to show this hold by induction?

Show that $$\frac3{1\cdot2\cdot4}+\frac4{2\cdot3\cdot5}+\dots+\frac{n+2}{n(n+1)(n+3)}=\frac16\left[\frac{29}6-\frac4{n+1}-\frac1{n+2}-\frac1{n+3}\right],\text{ for $n\in\mathbb N$}.$$ I try ...
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103 views

geomtry and induction Discrete Math

Claim: Suppose that we draw any number of straight lines in the plane, with the restriction that no two are parallel and no intersection point belongs to more than two lines. The lines divide up the ...
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57 views

Writing down the induction principle formally

Can one write principle of mathematical induction formally in the following way ($ P $ and $ S $ are a predicate and the successor function, respectively)? $$(\exists x\in\mathbb {N}(P (x))\wedge ...
2
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85 views

Checking an inductive proof on a combinatorial product

Consider the following product, for $n, k, i \in \Bbb Z_+, k \geq 2$: $$ {\prod_{\ell = 1}^i {n + k - \ell \choose k } \over \prod_{\ell = 1}^{i-1} { k + \ell \choose k}} \tag{$*$} $$ It has been my ...
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68 views

Proof for a finite number of elements

if I want to proof something for a restricted finite number of elements, meaning the following: Imagine that I have a theorem that is somehow similar to the following: For each element in ...
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180 views

Group-theoretic proof that an increasing multiplicative function is exponential

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $$\gcd(m,n)=1$$ Prove that ...
2
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231 views

Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
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20 views

Prove that if there are $2n$ points and $n^2+1$ straight lines connecting them, then there are at least $n$ triangles in this shape.

Proof by induction. For $n=2$, it says that if we have $2(2)=4$ points and $2^2+1=5$ lines connecting them to each other, then there are at least 2 triangles in this shape. Which is true (shown ...
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12 views

Induction for quantified statement with two discrete parameters

Given a quantified statement ∀n, n>0 (∃x, x>2k | x=2k+n) ( a subset of the natural numbers) This can logically this can be deduced as valid; however, I wish to use induction. Specifically I would ...
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28 views

Game of writing a binary sequence proof

Let $n \gt 2$ be a natural number. We consider the following game. Two players write a sequence of $0$s and $1$s. They start with an empty line and alternate their moves. In each move, a player writes ...
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28 views

Harmonic numbers, proof that h2^k >= 1+(k/2) with induction

I'm just starting with the concept of proving mathematical statements with induction. The complete exercise with solution can be found under: ...
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28 views

Prove that $10 | (n^a - n^b)$.

$n$ is a positive integer. Prove that there exists positive integers $a$ and $b$, $(a > b)$ such that $10 | (n^a - n^b)$. I have tried to prove this by induction on $n$, but I get stuck at the ...
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27 views

RE : Is greatest common divisor of two numbers really their smallest linear combination?

This is in reference to the same proof given in the post Is greatest common divisor of two numbers really their smallest linear combination? I couldn't add a comment there so I'm asking it here. I ...
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48 views

Number or regions formed when $n$ points on a circle are joined

The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$. This is a fact that I have read in several ...
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38 views

Mathematical Induction - Can terms be split up individually?

My doubt is regarding a particular technique of solving recurrence relations. Say, we have T(n) = T(floor(2n/5)) + 4T(floor(n/8)) + cn I wonder if we can find the individual time complexities of ...
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37 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
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55 views

Prove The following geometrically

$1^3 + 2^3 + 3^3 .......... +n^3 = (\frac{n(n+1)}{2})^2$ How am I supposed to prove this geometrically? I am sure we have to consider 3 dimensional space. But cannot figure it out how. Help
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44 views

A sequence defined as $a(n)=n-a(a(n-1))$ $n\geq 1,\ a(0)=0$, how to prove that $a(n)=⌊(n+1)(-1+√5)/2⌋$

$a(n)=n-a(a(n-1)), \ n \geq 1,\ a(0)=0$, to prove that $$ a(n)=⌊(n+1)\cdot \frac{\sqrt{5} - 1}{2}⌋. $$ This is an exercise of "Discrete Mathematics and Its Application".(Supplementary exercise 72 of ...
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67 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
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42 views

Taylor's Approximation proof by induction

I'm really stuck on this question Let $f(x)$ be a function on $[a,b]$ which is $(n + 1)$ times differentiable at every point, for some natural number $n$. Let $c$ be a point in $(a,b)$. Using ...
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43 views

Use of mathematical induction to compute $\int_0^{\pi/2} \sin^{2n+1}xdx$?

We are given a reduction formula for $\int \sin^n xdx$, namely $$\int \sin^nxdx = -\frac{1}{n}\sin^{n-1}x\cos x + \frac{n-1}{n}\int \sin^{n-2}xdx.$$ I know how to derive this reduction formula. How ...
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131 views

Variation of Nim: Player who takes last match loses

Here is a homework problem I can't understand the solution to. Can anyone help me understand why they are using "mod 4"? Can someone help me understand this strong induction example? Thanks everyone! ...
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51 views

Inductive proof about Jensen's inequality

The base case is easy. For the inductive step, i take $\lambda$ and $x$ to be as given, and then when I consider $f(\lambda_1 x_1 + . . . + \lambda_n x_n + \lambda_{n+1} x_{n+1})$ I get this is ...
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60 views

Proof by induction on contraction mapping?

Let $k:[0,1] \times [0,1] \to \mathbb{R}$ be continous, and $x(t) = \int_0^t k(t,s)x(s)ds$ for $0 \leq t \leq 1$. Not let $Tx(t) = \int_0^t k(t,s)x(s)ds$ and suppose $sup_{0 \leq t, s \leq 1}|k(t,s)|= ...
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367 views

Can I use Strong Induction to prove graph theory? - Hamilton Path/Tournament Graphs

I am working on the below proof. I am still new to proves so I am wondering if you guys can answer the following questions: 1) Did I use the inductive hypothesis correctly here? (By the inductive ...
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217 views

How to prove this equation using Mathematical Induction?

I was trying to prove this. I tried somehow but didn't get any idea. I think we can prove this using induction. I'd really appreciate it if you could help me. ...
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52 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
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31 views

induction on power elements in a group

Exercise: Prove by induction the following facts about power elements in a group. For all integers $k$ and $l$, $(a^k)^{l} = a^{kl}$ I am having trouble with induction on two integer variables. ...
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83 views

Show $0 \leq e^{-x} - \left( 1 - \frac{x}{n} \right)^n \leq \frac{x^2e^{-x}}{n}$ by induction

Show that if $0\leq x < n, n \geq 1$, and $n\in\mathbb{N}$ then $$ 0 \leq e^{-x} - \left( 1 - \frac{x}{n} \right)^n \leq \frac{x^2e^{-x}}{n}. $$ By using induction. Progress: Decided to split ...
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27 views

There are 10 non distinguishable balls and one of these has different weight

There are 10 non distinguishable balls and one of these has different weight(one does not know whether it is weigh more or less than others). One can use scales 3 times to compare their weight. You ...