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 ...

learn more… | top users | synonyms

1
vote
1answer
37 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 ...
3
votes
1answer
67 views

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 ...
0
votes
1answer
29 views

Induction proof involving sets

Suppose $A_1,A_2,...A_n$ are sets in some universal set $U$, and $n\geq2$. Prove that $\overline{A_1 \cup A_2 \cup ... \cup A_n}$ = $\overline{A_1} \cap \overline{A_2} \cap ... \cap \overline{A_n}$
0
votes
2answers
43 views

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
1
vote
1answer
58 views

Proof by induction or strong induction [on hold]

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 $n\in\mathbb{N}$.
0
votes
3answers
65 views

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 ...
2
votes
1answer
72 views

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 ...
2
votes
2answers
21 views

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 ...
4
votes
2answers
219 views

Prove the principle of mathematical induction in $\sf ZFC $

How does one prove the principle of mathematical induction using the standard axioms of $\sf ZFC $?
1
vote
1answer
235 views

Combinatorics - Catalan numbers recursive function proof by induction help

I'm trying to prove that $$C_n = \sum_{i=0}^{n-1} C_iC_{n-i-1}$$ when $1\leq n$ and $C_0 = 1$ by induction. ($C_n$ is the $n$th catalan number). For $n=1$ this is true. Assume it is true for $n=k$: ...
0
votes
3answers
65 views

Prove by induction that $2^m-1>m^3$ [on hold]

I have to prove that this inequality, for $m \geq 10$, using induction on $m$. Can anyone help me?
7
votes
10answers
1k views

Prove by induction that $5^n - 1$ is divisible by $4$.

Prove by induction that $5^n - 1$ is divisible by $4$. How should I use induction in this problem. Do you have any hints for solving this problem? Thank you so much.
1
vote
2answers
42 views

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 ...
3
votes
2answers
81 views

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 ...
0
votes
3answers
57 views
+50

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 ...
4
votes
1answer
58 views

how to solve this elementary induction proof

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},\ ...
0
votes
1answer
120 views

Proof completion: if $Y$ is a closed term in strong nf, then $Yx$ weakly reduces to a strong nf $Z$

I am self-studying Hindley & Seldin's Lambda-Calculus and Combinators. I would appreciate some help with filling in a final detail for a proof for the following statement regarding combinatory ...
14
votes
3answers
1k views

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 ...
3
votes
0answers
66 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 $\phi$ ...
4
votes
5answers
542 views

Proving that $f(n)$ is an integer using mathematical induction

I want to prove that $$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$$ is an integer for every integer $n \geq 1$. I define P(n) to be: $$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$$ is an integer. ...
7
votes
4answers
268 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 ...
0
votes
1answer
39 views

Inductive proof of inequality $a\le ab$ for nonnegative integers

I reading about of proof of the claim "If $a \ge 0$ and $b > 0$, then $a \le ab$. (Here $a$ and $b$ are integers.) The proof the author is employing is inductive. I understand the basis case; ...
1
vote
0answers
57 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 ...
0
votes
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 ...
0
votes
2answers
42 views

Prove that $n<(3/2)^n$ for any $n$ with induction [closed]

need help with induction with inequality, I suck at it. $n<\left(\frac{3}{2}\right)^n$ for any $n$
-1
votes
1answer
21 views

Inductive Proof Algorithm

so I'm working on an algorithms assignment and am having a tough time understanding what to do: The equation is: $$T(n) = 2T(n/4) + n = \Theta(n) = O(n)$$ Right now I have gotten this far: $$T(1) = ...
3
votes
3answers
41 views

Prove that $2n+1 \leq 2^n$ for $n \geq 3$ using mathematical induction.

Question: $2n+1 \leq 2^n$, for all $n \geq 3$ I've tried: Basis: $P(3) = 7 \leq 8 $, so basis step is valid Pick an arbitrary value from the universe, $k \geq 3$ Inductive Step: $2k + 1 \leq ...
1
vote
2answers
54 views

How can I show that $n^{n+2}<(2n)!$ for any integer $n$.

When I was try to show that the series $\sum_n \frac{n^n}{(2n)!}$ is convergent using comparison test, I stuck at the point $n^{n+2}<(2n)!$ I think it can be show using mathematical induction. If ...
3
votes
5answers
576 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 ...
6
votes
3answers
140 views

Math Induction Proof: $(1+\frac1n)^n < n$

So I have to prove: For each natural number greater than or equal to 3, $$(1+\frac1n)^n<n$$ My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3<3$$ $$\left(\frac43\right)^3<3$$ ...
2
votes
1answer
32 views

is my induction proof sufficient?

question; prove that $\forall\ n\ge4, n\in \mathbb{Z}, \ n!\gt n^2$. my work; let $n=4$ then $4!=24 \gt 4^2=16.$ true. now assume $n! \gt n^2$ is true for all $n\le k$ so now assume $k! \gt ...
1
vote
2answers
36 views

graph theory: show that for k=4 hesse diagram is not a planar graph

In this picture you can see the hesse diagrem of $\subseteq$ over $P(\{x,y,z\})$ For the set $A$ with $k$ elements, $k>0$ look at the diagram as a graph, it's vertices are the members of $P(A)$ ...
0
votes
2answers
30 views

Discrete Maths - Induction

I am having difficulty answering the following question: Can anyone show me how to solve this? I understand that I should be putting in a + 1 somewhere to simulate the next step, but I'm not sure ...
0
votes
2answers
66 views

Mathematical Induction Problem with Fraction

$$(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$$ I can't seem to solve it out to the point where I can prove it right or wrong. I always hit some sort of roadblock where I don't have enough info to prove it ...
4
votes
2answers
34 views

Proving Induction $(1\cdot2\cdot3)+(2\cdot3\cdot4)+…+k(k+1)(k+2)=k(k+1)(k+2)(k+3)/4$

I need a little help with the algebra portion of the proof by induction. Here's what I have: Basis Step: $P(1)=1(1+1)(1+2)=6=1(1+1)(1+2)(1+3)/4=6$ - Proven Induction Step: ...
2
votes
1answer
28 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
1
vote
1answer
41 views

scheme for n-dimensional induction

In slides: http://www.mathdb.org/notes_download/elementary/algebra/ae_A2.pdf I read the scheme for 2-dimensional induction, but Exists an scheme for n-dimensional induction? Thanks in advance!
0
votes
1answer
66 views

Spivak Chapter 2 Question 1 (i)

I don't understand Spivak's proof by induction of this exercise: Prove by induction $$1^2 + \ldots + n^2 = {n(n+1)(2n+1))\over 6}$$ It's true for $n = 1$ Then the proof continues adding $(k+1)^2$ ...
0
votes
1answer
38 views

a finite induction question from burton's elementary number theory

this question comes from burton's elementary number theory, 4th edition. question 3 in 1.1 says to use the second principle of finite induction to establish that $$for\ all\ n\ge1,\ ^{(a)}\ ...
2
votes
6answers
241 views

Discrete math induction problem.

I am stuck at this step in the inductive process and I was wondering if someone can help me out from where I am stuck. Question: if $n$ is a positive integer, prove that, ...
3
votes
2answers
152 views

How Many Miles to Retrieve an Object N Miles into a Desert?

The problem: Suppose that you are interested in retrieving an object located in the middle of the desert, n kilometers away. Your car can carry enough fuel to travel 3 kilometers, and you have an ...
1
vote
1answer
84 views

How to proof (using by mathematical induction)($n\in \mathbb{N}$) [closed]

I would appreciate it if somebody could help me with the following problem: Q: How to proof (using by mathematical induction)($n=2,3,4,\cdots$) ...
2
votes
1answer
54 views

Use principle of mathematical induction to show a function defined recursively is uniquely determined.

I'm having difficulty with the following taken from "Elementary Number Theory And Its Applications" by Rosen section 1.1 questions. "Use the Principal Of Mathematical Induction to show that the value ...
0
votes
2answers
43 views

Induction inequality check

check my proof, I feel like I made a mistake :) so I'm looking to prove that when $p(n)$ is $n!<n^n$, $p(n)$ is true for all $n>1$. Base Case $$ p(2) \iff 2!<2^2 \iff 2<4 $$ Assume p(k) ...
3
votes
2answers
99 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
0
votes
1answer
89 views

Strong Induction Base Case

Is a base case needed ? In response to many questions on this subject I offer the clarification below.
2
votes
1answer
20 views

Proof Concerning Linear Independence And Maximal Subsets

Serge Lang's Linear Algebra has, in chapter 1, a proof which seems rather long-winded. He wants to prove the following theorem: Theorem 3.1. let V be a vector space over the field K. Let ...
0
votes
1answer
33 views

Proof by induction and inequalities

I am stuck on this question: given $a_1a_2≤(\frac{a_1+a_2}{2})^2$ prove by induction of m that $$a_1a_2...a_p≤(\frac{a_1+a_2+...+a_p}{p})^p$$ where $a_i$ are all positive and real and $p=2^m$ (an ...
0
votes
2answers
43 views

Vacuous truth and (simple and complete) induction

The way I understand complete induction, as applied to the naturals at least, the inductive step consists of assuming that a given proposition $p_i$ is true for $1 \le i \le n$, and from this deduce ...
2
votes
1answer
42 views

Generalized Induction Verification

Consider the following simple exercise. Prove or disprove: $\gcd(km, kn) = k \gcd(m, n)$, where $m, n, k$ are natural numbers. Now, this is easy to prove using prime factorization. Knowing that ...