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
0answers
24 views

coin problem with two coins, inductive proof

I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime factors in common. I will write $$ ...
1
vote
2answers
35 views

Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction

I need help with finding the formula and proving it by induction. Am stuck, but the professor says we should know this by now.
1
vote
1answer
788 views

Sets induction problem (complement of intersection equals union of complements)

Let $n\ge 2$ and $A_1,\dots,A_n$ be sets in some universe $S$. In this problem we will give a proof by induction of the identity $$\left(\bigcap_{i=1}^nA_i\right)^c=\bigcup_{i=1}^nA_i^c\;.$$ ...
0
votes
3answers
94 views

Number Theory - Proof by Induction

Show that: $2903^n - 803^n - 464^n + 261^n$ is divisible by $1897$ for all integers $n\geq1$ using induction.
2
votes
1answer
39 views

Reference for $F$-algebras and induction?

I've been learning about $F$-coalgebras and coinduction from this fantastic paper, which has really helped me get a feel with its many examples. I'm starting to struggle with reconciling the ...
3
votes
1answer
26 views

Direct proof of principle of transfinite induction

This is a problem from the book Set theory by You-Feng Lin. Principle of Transfinite Induction Let $(A,\le)$ be a well-ordered set. For each $x \in A$, let $p(x)$ be a statement about $x$. If for ...
-1
votes
0answers
33 views

Can anyone show this question? [on hold]

show that if we consider the subset of the integers {S = k, k +1, k +2, ...} and T a subset of S such that k belongs to T and (k, k +1, ..., n ⇒ T 1 + T n), then T = O.
-4
votes
1answer
30 views

Show this question please: [on hold]

Show that if we consider the subset of integers S={k, k+1, k+2,...} and T a subset of S such that K belongs to T and (n belongs to T and (n+1) belongs to T) so T=S.
4
votes
3answers
234 views

Proving $n! > n$ for $n > 2$ using mathematical induction

I have to prove $n<n!$ for all $n>2$ by mathematical induction. I did it as follows. I proved the base case. Then let it be true for $K>2$: $$ K<K! $$ I have to prove, $$ ...
2
votes
2answers
105 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
1
vote
1answer
56 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
2
votes
3answers
40 views

How to solve this recursive equation?

I've got this recursive equation: $$ T(n) = \begin{cases} 2, & \text{if $n = 2$} \\ 2T(n/2) + n, & \text{if $n = 2^k$ where k > 1, $k \in \mathbb{N} $} \\ \end{cases} $$ I know I should ...
-1
votes
3answers
45 views

how to prove $1/n (1-(1/2)^n)$ decreasing without using differentiation

$a(n)=1/n (1-(1/2)^n)$ prove $a(n+1)<a(n)$ for n>0 by differentiating slope comes negative and then we can prove it . but i wanted to solve it without that . can someone help
2
votes
0answers
33 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) = ...
5
votes
5answers
183 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
1
vote
1answer
51 views

What is wrong with the following induction argument?

I found a problem on a note on induction. The problem went like this: "Let $n$ be a non-negative integer. Suppose we are given a triangle and n points inside it, with no three of the given $n + 3$ ...
-1
votes
0answers
27 views

Induction method(theory of computation). [on hold]

Prove by induction that |A*B|=|A|*|B|,for all set of A&B.
0
votes
1answer
26 views

Induction Problem Number of Tiles on Floor

I took a discrete math course about a year ago, and I recently decided to crack open my book again as a refresher on induction proofs and problems. I ran across this problem, which I didn't remember ...
0
votes
2answers
40 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
32
votes
5answers
3k views

Prove that the 25 people can be seated in this way

5 mathematicians, 5 biologists, 5 chemists, 5 physicists, and 5 economists sit around a large round table. Prove that the 25 people can be seated such that, if A and B are two different people with ...
-1
votes
1answer
56 views

Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
1
vote
3answers
55 views

Showing that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$

I'm trying to prove that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$. I figured induction was the way to go, so I tried. This is what I've accomplished so far: Proved ...
1
vote
1answer
42 views

Maximum and average number of inversions in array by induction

Just for your information, an inversion in an array $a$ is any ordered pair of points $(i, j)$ where $i < j$ and $a_i > a_j$. I can prove the maximum and average number of inversions in an ...
21
votes
5answers
2k 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 ...
0
votes
0answers
12 views

Math Bases of Comparison and Association [closed]

My question is about the cognitive phenomenon of intuitive pattern-matching or association by similarity / dissimilarity. Imagine a situation where a person has a particular experience, which might ...
3
votes
2answers
69 views

Limit of a sequence of averages (three variables)

Let $a_0 = 0$, $a_1 = 0$, $a_2=1$ and for $n>2$, $a_n = \dfrac{a_{n-1}+a_{n-2}+a_{n-3}}{3}$. Consider $\lim\limits_{n \to +\infty} a_n$. Using a python script I found that $a_n$ tends to ...
0
votes
1answer
42 views

Prove if n<m there is at least one [(n/m)]?

Suppose there are n programmers in m cubicles. Prove that there must be at least one cubicle containing at least [(n/m)] programmers. Note: I was not able to find the right sign [ is returning first ...
1
vote
1answer
29 views

For every $n$ there exists $m$ such that $m/n$ is an upper bound but $(m-1)/n$ is not

This is a problem discussed in Analysis 1 by Terence Tao. $E$ is a non-empty set of Real numbers. $ n\geq 1$, $L,K$ are two integers such that $L<K$. Let $\frac{L}{n}$ is not an upper bound of ...
3
votes
5answers
311 views

What does “Prove by induction” mean?

What does "Prove by induction" mean ? I've heard it a lot! Would you mind giving me an example? Thanks
2
votes
4answers
72 views

Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$? [duplicate]

I have tried this question so hard but still stuck here. It seems like easily provable if all $n$ are all positive numbers but in this question, the $n$ is bigger than $1$. original question : prove ...
1
vote
1answer
52 views

Proof of equation $\sum_{k}{n\brack k}a_k = n!2^{n-1}$ by induction

I'm trying to prove to following equation: $$\sum_{k=0}^{n}{n\brack k} a_k = n!2^{n-1};\ \ \ n\ge 1$$ $a_n$ - number of ordered partition of set. We have following recursion dependencies: $a_n = ...
2
votes
2answers
68 views

Show that $(n+1)^{n+1}>(n+2)^n$ for all positive integers

Show that: $(n+1)^{n+1}>(n+2)^n$ holds for all positive integers I tried using induction: for $n=1$ we have 4>3 then for $n+1$ we have to show that $(n+2)^{n+2}>(n+3)^{n+1}$ and here I ...
1
vote
1answer
60 views

Induction on prime numbers

To dive straight into the question: is there a form of induction which works on prime numbers? I've thought, and while I'm pretty sure it can be done om numbers such as even numbers or numbers ...
0
votes
1answer
32 views

how to prove using induction that sum of terms?

Prove that $\displaystyle\sum\limits_{i=1}^{k}\left(\dfrac{1}{(2i-1)}\dfrac{1}{(2i+1)}\right) = \dfrac{k}{(2k+1)}‎‎$ My Base of Induction is to check that it is true for i=1, so: ...
0
votes
1answer
30 views

Readings on more general/abstract notions of induction related to logic

Can someone suggest references to understand the more general/abstract concept of induction? Specifically, I am trying to justify to myself what is called induction on the "complexity of a ...
2
votes
2answers
87 views

Proof of equality $\sum_{k=0}^{m}k^n = \sum_{k=0}^{n}k!{m+1\choose k+1} \left\{^n_k \right\} $ by induction

I have a problem with following equality: $$\sum_{k=0}^{m}k^n = \sum_{k=0}^{n}k!{m+1\choose k+1} \left\{^n_k \right\} $$ And I would like to use induction in following way: Base: $$ m = n $$ And: $$ ...
1
vote
1answer
260 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 $n=k$: ...
3
votes
1answer
72 views

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 ...
1
vote
1answer
28 views

Induction proof for continued fractions

Recently while preparing for a maths test, I got this question in a book: Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms. Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
-2
votes
2answers
59 views

What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
0
votes
2answers
25 views

Help me find the wrong in this inductive method proof?

Problem: Prove that: In a classroom with n student, if there is a girl student, all students of this class are girl. Solving: Let f(n) is the clause: In the class, if there is 1 girl student, all of ...
2
votes
1answer
63 views

Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?

The "principle of mathematical induction" says that for a subset $S$ of $\omega$ (where $\omega$ is the set of all natural numbers), if $0 \in S$ and $n \in S \implies n^+ \in S$, then $S = \omega$. ...
2
votes
5answers
111 views

Proof via induction $1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$

(b) Prove that for every integer $n \ge 1$, $$1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$$ This is the second part of a two part question. Part (a) was the following: ...
0
votes
1answer
30 views

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 ...
0
votes
0answers
38 views

Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
4
votes
2answers
110 views

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: ...
2
votes
2answers
36 views

Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$

I have to prove the following by induction: $$(1+x)^n \le 1+(2^n-1)x$$ for $n \ge 1$ and $0 \le x \le 1$. I start by showing that it's true for $n=1$ and assume it is true for one $n$. ...
1
vote
2answers
53 views

How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
3
votes
2answers
592 views

Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
3
votes
3answers
76 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...