For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the *base case*, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant ...

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364 views

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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235 views
+100

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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343 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
4
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70 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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96 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 ...
4
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102 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 $n$) ...
4
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88 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 ...
4
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183 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, ...
4
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115 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 ...
3
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42 views

Induction Can't Prove Complexity?

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(x) > L$ where $K(x)$ is the Kolmogorov complexity of natural number $x$ and $L$ is a sufficiently ...
3
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22 views

Proof of Leibniz formula from Laplace expansion

I'm trying to prove Leibniz formula for the determinant using Laplace expansion. Here's my attempt: For a $1 \times 1$ matrix $A = \begin{pmatrix}a_{11}\end{pmatrix}$, define $\det A = a_{11}$. For ...
3
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39 views

Induction proof - not sure how to proceed with next step

Define two sequences $A_n, B_n$ as follows: \begin{align*} A_1 &= 1\\ A_2 &= 3\\ A_3 &= 2 \cdot 3+1=7 \\ A_4 &= 2 \cdot 7 + 3 = 17\\ A_5 &= 2 \cdot 17 + 7 = 41\\ A_n &= ...
3
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48 views

Verify my induction proof for Balanced Ternary expressions

In first place I apologize if you find a grammatical error, my English is not too good for now, but I'm work on it. That also goes for errors in my question (this is my first post). I encounter the ...
3
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0answers
60 views

Using induction to prove a congruence?

Let $a = 2+\sqrt{3}.$ By analogy to complex numbers let R$(a)$ be $r,$ the non-surd part of $r + s\sqrt{3}.$ I would like to show that a necessary but by no means sufficient condition that ...
3
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103 views

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

For natural numbers $n\ge1$ show the following inequality using induction. $$n^{1/n}\le 1+\sqrt{\frac{2}{n}}$$
3
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62 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)! ...
3
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104 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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565 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 ...
3
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217 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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22 views

How to find variance of k+1 elements if variance of k elements is known?

I need to find the variance of k+1 elements given the variance of k elements. I can also store other features for k elements like mean ($\mu_n$) etc. So, given the below function's value, $$ ...
2
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62 views

Properties of Ackermann's function

I want to show the following properties of Ackermann's function: $A(x,y)>y$. $A(x,y+1)>A(x,y)$. If $y_2>y_1$, then $A(x,y_2)>A(x,y_1)$. $A(x+1, y) \geq A(x,y+1)$. $A(x,y)>x$. If ...
2
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65 views

Trouble with induction on the length of a word

In the accepted solution of the question If L is regular, prove that $\sqrt{L}=\{w:ww\in L\}$ is regular the answerer made the claim that "What's left is to show that $δ ′ (q_{0}' ,w)=h$ , which can ...
2
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27 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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68 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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90 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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154 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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163 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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51 views

Proving the base case for a problem in elementary number theory

I have a question about how to prove statements such as the following, using induction: If $p \mid a_1a_2 \cdots a_k$, then $p \mid a_i$ for some $i$, $i = 1, 2, \ldots, k$, where $p$ is prime. ...
2
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86 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 ...
2
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101 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 $1, 2, \ldots , 9$). ...
2
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47 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 ...
2
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144 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 ...
2
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58 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 ...
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100 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 ...
2
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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 ...
2
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289 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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28 views

Proving recursive formula via induction leads to extra term?

I have been asked the following question, and despite spending the last 30 minutes on it, have not come up with a good result: Define f(1) = 2, and f(n) = f(n-1) + 2n for all n ≥ 2. Find a ...
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50 views

Is it possible to use induction to prove Laplace Expansion Theorem?

Laplace expansion theorem is used to find the determinant of an $n \times n$ matrix. It can be applied along a row or along a column. Let's assume that we can prove this theorem using induction (As it ...
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48 views

Verification of Basic Proof in Spivak Calculus (Induction)

I have began working through Spivak's Calculus book and trying to do the problems at the end of the chapters. I am rather new to proof, so forgive the naivety of this type of question. I am wanting ...
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31 views

Induction Proof from Thomas Judson book on abstract algebra

I'm trying to prove $$^n\sqrt{a_1\times a_2\times...\times a_n}\leq \frac{1}{n}\sum_{k=1}^na_k, \quad a_i\in \mathbb{Z}^+$$ by Induction. The case is true for $n=1$ so I assumed true for $n=k$. I then ...
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66 views

Using induction to prove the “hockey stick theorem”

The question we were given was (where $^nC_c$ is $n$ choose $c$): Show, using induction and the fact that $^nC_c + ^nC_{(c+1)} = ~^{(n+1)}C_{(c+1)}$, the "hockey stick theorem": the sum from $k=c$ ...
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20 views

Show that $f(b^i n) \le c^i f(n)$

Let $f$ be a b-smooth function. Let $c$ and $n_0$ be constants such that $f(b n) \le c f(n)$ $\forall $ $n \ge n_0$. Show that $\forall $ $ i \in \mathbb{N}, f(b^i n) \le c^i f(n)$ I thought I should ...
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45 views

Induction proof: $\det(M) = \prod_{1 \le j \le n} (x_j - x_i)$

Following problem: Let $\mathbb{K}$ be a Field and $M = \begin{pmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & ...
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19 views

Recurence with multiple variables and functions

Is there an easy way to solve a recurrence given with two variables and three different functions? Actually I'm looking for the solution of: $$A(n,k)=A(n-2,k-1)+A(n-3,k-1)+R(n-2,k-1)+L(n-2,k-1) $$ ...
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37 views

Exercise in induction (including double indices)

The following is for an exam preparation exercise in induction Problem: Let $f(x)=|x| = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}$ and $g(x)=|x|^{2k+1}$. Let $N \leq n$ and let $k \in \mathbb{Z}$, ...
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30 views

Generalized Bernoulli's inequality of the form $\frac{1}{1-\sum_{i=1}^nx_i}\geq\prod_{i=1}^n(1+x_i)$

We got to prove the generalized Bernoulli's inequality given below for $x_1,...,x_n\geq0$ using induction. $$ \frac{1}{1-\sum_{i=1}^nx_i}\geq\prod_{i=1}^n(1+x_i)\geq 1+\sum_{i=1}^nx_i$$ The right ...
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196 views

Proving a recursive algorithm as correct using induction

My objective is to give a recursive algorithm for finding the maximum of a finite set of integers, "making use of the fact that the maximum of n integers is the larger of the last integer in the list ...
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70 views

What really is inductive reasoning?

I am looking for concrete information on "induction" in the old sense of the word. Now it is used to refer to proof by "mathematical induction"; however, I am aware that it also means inductive ...
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22 views

Proving that step-wise function is $\mathcal C^{\infty}$

I'd like a proof verification of the following, please. I want to prove that $H: \Bbb R \to \Bbb R$ is of class $\mathcal C^{\infty}$, with $H$ defined by: $$H(x) = \left\{\begin{align} &h(x) = ...
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33 views

How do I solve this recurrence relation and prove by induction?

I have this summation formula: $T(n)=\sum_{i = 1}^{n}T(n-i)T(i-1)$. Base case is $T(0)=1$, $T(1)=1$. How do I find the recurrence relation and prove it by induction?