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 ...

learn more… | top users | synonyms

86
votes
6answers
9k views

Induction on Real Numbers

One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far. Of course you have to change ...
76
votes
9answers
12k views

Surprise exam paradox?

I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows: A maths teacher says to the class ...
53
votes
10answers
2k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
51
votes
11answers
6k views

All natural numbers are equal.

I saw the following "theorem" and its "proof". I can't explain well why the argument is wrong. Could you give me clear explanation so that kids can understand. Theorem: All natural numbers are equal....
48
votes
16answers
4k views

How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
47
votes
2answers
3k views

Proof of 1 = 0 by Mathematical Induction on Limits?

I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. Define $P(m)$ to be the statement: $\quad \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\...
46
votes
12answers
7k views

Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
42
votes
7answers
5k views

Dominoes and induction, or how does induction work?

I've never really understood why math induction is supposed to work. You have these 3 steps: Prove true for base case (n=0 or 1 or whatever) Assume true for n=k. Call this the induction hypothesis....
40
votes
9answers
7k views

Good examples of double induction

I'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" ...
38
votes
15answers
4k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ \...
38
votes
19answers
9k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
37
votes
26answers
6k views

How to teach mathematical induction?

Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of ...
36
votes
1answer
50k views

What's the difference between simple induction and strong induction?

I just started to learn induction in my first year course. I'm having a difficult time grasping the concept. I believe I understand the basics but could someone summarize simple induction and strong ...
34
votes
5answers
4k 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 ...
34
votes
3answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
33
votes
10answers
10k views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
32
votes
3answers
3k views

Can't find the flaw in the reasoning for this proof by induction?

I was looking over this problem and I'm not sure what's wrong with this proof by induction. Here is the question: Find the flaw in this induction proof. Claim $3n=0$ for all $n\ge 0$. ...
32
votes
14answers
5k views

Why doesn't mathematical induction work backwards or with increments other than 1?

From my understanding of my topic, if a statement is true for $n = 1,$ and you assume a statement is true for arbitrary integer $k$ and show that the statement is also true for $k + 1,$ then you prove ...
32
votes
7answers
2k views

Must we use induction to prove a statement for all integers

This question is prompted by a remark from Bill Dubuque in his answer to this question on proving a particular sum without using mathematical induction. From Bill's answer: A proof that a ...
31
votes
7answers
2k views

We all use mathematical induction to prove results, but is there a proof of mathematical induction itself?

I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like $$1+2+3+\cdots+n = \frac{n(n+1)}...
28
votes
14answers
954 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
28
votes
1answer
396 views

Uses of “Collatz induction”?

The Collatz conjecture is equivalent to the following "induction principle": If $P(0) \land P(1) \land (\forall{x} P(3 \cdot x + 2) \implies P(2 \cdot x + 1)) \land (\forall x P(x) \implies P(2 \cdot ...
27
votes
7answers
8k views

Is there no solution to the blue-eyed islander puzzle?

Text below copied from here The Blue-Eyed Islander problem is one of my favorites. You can read about it here on Terry Tao's website, along with some discussion. I'll copy the problem here as ...
26
votes
9answers
6k views

Why $a^n - b^n$ is divisible by $a-b$?

I did some mathematical induction problems on divisibility $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5. Can these be generalized as $a^n$ $...
26
votes
4answers
2k views

Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the ...
24
votes
6answers
1k views

IMO 1987 - function

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
23
votes
5answers
3k 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 ...
23
votes
5answers
1k views

How does one actually show from associativity that one can drop parentheses?

I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this? $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$...
22
votes
3answers
699 views

A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...
22
votes
7answers
693 views

Illustrative examples of a phenomenon in the logic of mathematical induction

It is said (and I myself have said) that in some cases the easiest way to prove a statement by mathematical induction is to prove a stronger statement by mathematical induction, because then one has a ...
21
votes
6answers
4k views

Why is mathematical induction a valid proof technique? [duplicate]

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over ...
20
votes
9answers
1k views

What is the purpose of the first test in an inductive proof?

Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$" Like this: Prove that $1+2+3+...+...
20
votes
5answers
260 views

Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those ...
19
votes
6answers
2k views

Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$

I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: $$0^2\binom{n}{0}+1^2\binom{n}{1}+2^2\binom{n}...
19
votes
6answers
4k views

Why doesn't induction extend to infinity? (re: Fourier series)

While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic ...
19
votes
7answers
1k views

What are some examples of induction where the base case is difficult but the inductive step is trivial?

According to Wikipedia: ...proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as n = 0 or n = 1 and ...
18
votes
3answers
1k views

Proof by induction that $n^3 + (n + 1)^3 + (n + 2)^3$ is a multiple of $9$. Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate $P$...
17
votes
12answers
5k views

Prove that $ n < 2^{n}$ for all natural numbers $n$.

Prove that $ n < 2^{n} $ for all natural numbers $n$. I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k<2^{k}$. Considering $k+1 <2^{k}+1$, ...
17
votes
9answers
2k views

There are no bearded men in the world - What goes wrong in this proof?

Several years ago in a textbook I read this example as a faulty use of proof by induction. I never really realized why it fails. Here it goes: Theorem. There are no bearded men in the world. ...
17
votes
0answers
406 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 $p(n)$ ...
16
votes
5answers
889 views

Fibonacci number identity.

How do I see that $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$, $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
16
votes
4answers
2k views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
15
votes
11answers
7k views

Prove that $1 + 4 + 7 + · · · + 3n − 2 = \frac {n(3n − 1)}{2}$

Prove that $$1 + 4 + 7 + · · · + 3n − 2 = \frac{n(3n − 1)} 2$$ for all positive integers $n$. Proof: $$1+4+7+\ldots +3(k+1)-2= \frac{(k + 1)[3(k+1)+1]}2$$ $$\frac{(k + 1)[3(k+1)+1]}2 + 3(k+1)-2$...
15
votes
3answers
2k views

Proof that $\pi$ is rational

I stumbled upon this proof of $\pi$ being rational (coincidentally, it's Pi Day). Of course I know that $\pi$ is irrational and there have been multiple proofs of this, but I can't seem to see a flaw ...
15
votes
6answers
1k views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
15
votes
2answers
10k views

Induction on two integer variables

Assume you want to prove an identity such as $$\sum_{k=m+1}^{n}A(k,m)-B(k,m)=S(m)+T(n,m)\qquad\text{for } n,m\in \mathbb{Z},n,m\geq 0.$$ Added: I applied mathematical induction on $m,n$ to prove it....
15
votes
6answers
792 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
15
votes
5answers
429 views

Proving $\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}}$ by induction for all $n> 2$.

I am trying to prove $$\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}} \text{ for all }n > 2.$$ Here is the original source (Problem 1B, on page 12 of PDF) Can this be proved by ...
15
votes
5answers
306 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
14
votes
12answers
21k views

Proof that $n^3+2n$ is divisible by $3$

I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs! Problem: For any natural number $n , n^3 + 2n$ is divisible by $3.$ This makes sense ...