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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Mathematical Induction - Inequality

Does anyone have any idea on how to complete the inductive step? Thm: For all $n >= 0~~~~ 6^n + 4 > n^3$ Pf: by Induction     Let $P(n)$ be proposition that $~6^n + 4 ...
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Prove by induction a formula for $x_{k+1}=\frac{x_k}{x_k+2}$, $x_1=1$

I have a IT Maths exam coming up and I just can't figure out this question. Any help would be appreciated thanks. A sequence of integers $x_1,x_2,\dots,x_k,\dots$ is defined recursively by ...
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induction exercise doubt

the exercise states: Let $x_1 , ...,x_n$ be strictly positive numbers such that their product is equal to 1. Show then that $\sum_{k=1}^{n} {x_k} \ge n $, for every $n \ge 2$. My solution: for the ...
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proof by mathematical induction that the total number of subsets of a set is $2^n$ [closed]

What is the proof by mathematical induction that the total number of subsets of a set is $2^n$?
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Proof in induction recrusive function

We've got the following function: $$f:N \rightarrow N$$ $$f(0) = 1$$ $$f(K+1) = (K+1)\times F(K)$$ How can I proof in induction the following: $$1\times f(1)+2\times f(2)+3\times f(3)+...+n\times ...
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Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...
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1answer
53 views

problem: Applying Well-Ordering Principle to prove a fact

problem: Prove the fact using WOP: every amount of postage that can be assembled using only 10 cent and 15 cent stamps is divisible by 5. The problem provides a template for this proof and asks that ...
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Use induction to prove trignometric identity with imaginary number

Prove by induction that if $i^2 = -1 $, then for every integer $n >= 1$, $[\cos(x) + i\sin(x)]^n = \cos(nx) + i\sin(nx)$. My solution so far: 1. It can be easily shown that it is true for n = 1. ...
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68 views

Introductory Induction Proof

I am in currently in a discrete mathematics class, and I've done well on every problem I've encountered. Unfortunately, I find myself weak at some of the seemingly straight forward induction problems. ...
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How to prove $\tan^{-1}(n+1)-\tan^{-1}(n-1)=\tan^{-1}\big(\frac{2}{n^2}\big)$?

Prove $$\tan^{-1}(n+1)-\tan^{-1}(n-1)=\tan^{-1}\big(\frac{2}{n^2}\big)$$ for $n \ge 1$ If I use mathematical induction how do I manipulate the numbers to fit in the induction hypothesis? Is there ...
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Induction on GCD problem [duplicate]

This is a two part question Given $\gcd(a,b) = 1$ consider $$\gcd \left( \frac{a^n - b^n }{a-b}, a- b\right) $$ It appears that the value of this is always equal to $n$ or $1$. How to prove it? ...
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How to complete a proof by induction

I was trying my hand at proof by induction and got this exercise from the first chapter of Wissam Raji's "An introduction in elementary number theory". I have to prove by induction that $n< 3^n ...
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4answers
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How to prove this $(n+1)^n < n^{n+1}$ for $\space n \ge 3$

I'm having some more trouble with induction I know how to prove this using $\ln$, but I need to use induction only. prove that: $(n+1)^n < n^{n+1}$ for any $ n\ge 3$
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1answer
55 views

Inductive proof of a formula for Fibonacci numbers

May someone help me? I am trying to use induction to prove that the formula for finding the $n$-th term of the Fibonacci sequence is: ...
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5answers
77 views

prove by induction $7 \mid 3^{3^n}+8$

Okay so ive been trying to prove this for about 5 hours... really need salvation from the geniouses around here. prove by induction $7\mid 3^{3^n}+8$ i really need some directions on what to do ...
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1answer
19 views

Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1". $J(2n) = 2J(n) - 1$, for $n \ge 1$ I don't quite ...
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73 views

Prove that $(2n+1)+(2n+3)+\dots +(4n-1) = 3n^2$ by induction

Note: This is for self study, the book is Elementary analysis by Kenneth. A. Ross How to prove the following by mathematical induction, I am stuck
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Induction inequality on sum of reciprocals

I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ ...
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1answer
26 views

Expanding Base 2B representation of an integer

Consider an integer$L$ written in Base 2B which digits $$a_n a_{n-1} a_{n-2} ... a_1 B$$ Where $a_i$ are arbitrary constants such that $9 \le a_i < 2B$. I am attempting to prove that the square ...
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Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
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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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Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
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1answer
37 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
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1answer
16 views

Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
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2answers
53 views

If $a_i>o$ then $(a_1a_2\cdots a_{2^n})^{1/2^n}\leq \frac{a_1+a_2+\cdots+a_{2^n}}{2^n}$

I need help to prove this inequality, I have no idea how to proceed with the inductive step: $$a_1,a_2,\ldots,a_{2^n}>0 \Longrightarrow(a_1a_2\cdots a_{2^n})^{1/2^n}\leq ...
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1answer
32 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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54 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
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Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
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Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
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1answer
31 views

Can someone explain the logic behind this step in a induction problem

There is a question in the book that I don't quite understand. Question Show that $n^2$ is smaller than $2^n$ whenever $n\ge5$. At the $k+1$ step it gets very whacked and confusing. $k+1$ ...
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1answer
39 views

Prove by Structural induction, circular permutations

Prove by Structural Induction: For a circular permutation of $n$ elements, the number of permutations is $(n-1)!$ How is this done?
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How do I reduce this induction problem at the k+1 step

Show that $2^n > n^2$ through induction and so far I got to the $k+1$ step, but I am stuck. I have $2^{k+1} = 2 +2^k$, but I don`t know how the book turned it into $k^2 +k^2$. The book then ...
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Better proof that $n \leq 2n$ for all natural numbers?

I tried proving via induction on naturals that $n \leq 2n$ for each natural $n$. Obviously, $0 \leq 2(0)$, and then assuming for any given $n$, $n \leq 2n$, you just show that $n + 1 \leq 2(n + 1).$ ...
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79 views

Proving inequality $3^{n^2} > (n!)^4$

Prove that $3^{n^2} > (n!)^4$ for all positive integers $n$. I tried to use induction on this problem but failed to do so. I instead tried to prove $3^{2n+1}>(n+1)^4$, but couldn't come up ...
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Proof by induction $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n [duplicate]

Prove by induction that $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n
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Induction proof involving sets

Suppose $A_1,A_2,...A_n$ are sets in some universal set $U$, and $n\geq2$. Prove that $\overline{A_1 \cup A_2 \cup ... \cup A_n}$ = $\overline{A_1} \cap \overline{A_2} \cap ... \cap \overline{A_n}$
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1answer
65 views

Proof by induction or strong induction [closed]

I have to prove the following using induction or strong induction Suppose $a\in\mathbb{Z}$. Prove that $5 | 2^na$ implies $5 | a$ for any $n\in\mathbb{N}$.
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Lightbulb puzzle related to proofs by induction.

The puzzle question was as follows: There is a circle of $n > 2$ lights with a switch next to each of them. Each switch can be flipped between two positions, thereby triggering the on/off states of ...
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1answer
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Mathematical Induction proof for $(n!)^2 > n^n$

I have the math problem (induction proof - $n!^2 > n^n$) that I try to solve and I haven't yet managed to get it right so maybe somebody could help me. My current plan solving the problem is the ...
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2answers
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Proving arithmetic series by induction

How do I prove this statement by the method of induction: $$ \sum_{r=1}^n [d + (r - 1)d] = \frac{n}{2}[2a + (n - 1)d] $$ I know that $d + (r - 1)d$ stands for $u_n$ in an arithmetic series, and the ...
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How to prove Post's Theorem by induction?

The proof of post's theorem is given in my textbook in two pages of explanation using a non-induction method. I was told that ,using induction on length of the proof, one can get a simpler and more ...
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Number of ways to write a natural as a sum of naturals [duplicate]

Problem: Let $n$ be a natural number, and $S(n)$ be the number of ways $n$ can be written as a sum of naturals. For instance, $S(3) = 4$ because $3 = 2+1 = 1+2 = 1+1+1$ and these are four different ...
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Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
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1answer
75 views

how to solve this elementary induction proof: $\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\le\ 2-\frac{1}{n}$

this is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction the question; $$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ ...
22
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5answers
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What makes induction a valid proof technique?

What makes induction (over natural numbers) a valid proof technique? Is $$ \dfrac{ P(0) \quad \forall i \in \mathbb{N}. P(i) \Rightarrow P(i+1) }{ \forall n \in \mathbb{N}. P(n)} $$ just taken for ...
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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 $\phi$ ...
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69 views

Infinite descent method and strong induction

I encountered the following statement of the infinite descent principle (PID): PID. Let $p(n)$, $n \in \mathrm{N}$, be an arbitrary property of natural number $n$. Assume that (e) $p(1)$ is ...
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59 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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1answer
57 views

am i cheating in this number theory proof?

the question (from burton's elementary number theory); $verify\ that\ \forall n\ge 1,$ $$2\cdot6\cdot10\cdots(4n-2)=\frac{(2n)!}{n!}$$ my work/proof; this is obviously true for $n=1$, so assume ...
7
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4answers
277 views

The largest integer less than $n$ is $n-1$

Let $n$ be a positive integer. Prove that the largest integer which is less than $n$ is $n-1$. Attempt of a solution: Since $n$ is a positive integer $n>0$. I think I have to use the well-ordering ...