Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [induction]

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 for questions about induction over natural numbers but is also appropriate for other kinds of induction such as transfinite, structural, double, backwards, etc.

Mathematical induction 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 relations which are well-founded, this tag is aimed mostly at questions about induction over natural numbers.

In general use, induction means inference from the particular to the general. This is used in terms such as inductive reasoning, which involves making an inference about the unknown based on some known sample. Mathematical induction is not true induction in this sense, but is rather a form of proof.

Induction over the natural numbers generally proceeds with a base case and an inductive step:

  • First prove the statement for the base case, which is usually $n=0$ or $n=1$.
  • Next, assume that the statement is true for an input $n$, and prove that it is true for the input $n+1$.

The following variant goes without a base case: Assuming the statement is true for all $n\in\mathbb N$ with $n < N$, prove that is true for $N$, too. This has to be done for all $N\in\mathbb N$.

10150 questions
2
votes
1 answer

Prove the statement by induction

Prove that ∀n ∈ N{1}, (1/1!)+(1/2!) + (1/3!) +···+ (1/n!) < 3− [2/(n+1)!] after suppose it is true for n = k, so (1/1!)+(1/2!) + (1/3!) +···+ [1/ (n)!]< 3− [2/(k+1)!] then for n= k+1, (1/1!)+(1/2!) + (1/3!) +···+ [1/ (k)!] + [1/(k+1)!]< 3−…
Haoran Chen
  • 169
2
votes
1 answer

Use Induction to show that when n circles divided the plane

Use Induction to show that when n circles divided the plane into regions, those regions can be colored into 2 different colors such that no regions with a common boundary are colored the same. My think: Let p(n) : "the statement, coloring regions…
AJGS
  • 23
2
votes
3 answers

Mathematical Induction (product of $n$ consecutive numbers)

Assumption: $$(n+1)(n+2) \cdots (2n) = (2^n)\cdot 1 \cdot 3 \cdot 5 \cdots (2n-1)$$ Prove for $n+1$: $$(n+2)(n+3) \cdots (2(n+1)) = (2^{n+1}) \cdot 1 \cdot 3 \cdot 5 \cdots (2(n+1)-1)$$ Using the assumption, I divide both sides by $(n+1)$ and…
meiryo
  • 875
2
votes
2 answers

Inductive question

Asked to prove that for $n \in \mathbb{Z^+}$: $\sqrt{1}+\sqrt{2}+...+\sqrt{n}\ge\frac{2}{3}n\sqrt{n}$ by mathematical induction. My inductive hypothesis is then: $\sqrt{1}+\sqrt{2}+...+\sqrt{k}\ge\frac{2}{3}k\sqrt{k}$ I've come to this…
Kurtooso
  • 148
2
votes
1 answer

Inductive proof of inequality involving summation and nested exponents

What is the best way to approach this problem? $$\sum_{k=1}^{n} \frac{1}{2^{k^2}} \leq 1-\frac{1}{2^{n^2}}$$ So far I have proved this works for a base case where $n=1$ $$\sum_{k=1}^{1} \frac{1}{2^{k^2}} \leq 1-\frac{1}{2^{1^2}}$$ $$\frac{1}{2}…
Thicc
  • 81
  • 1
  • 6
2
votes
3 answers

Prove that $n^4 \mod 8$ is identically equal to either 0 or 1, $\forall \ n\in \mathbb{N}$.

I feel pretty confident about the first half of my solution for this, however I don't like how I used the induction hypothesis on the case for even integers, it feels like it isn't doing anything useful since it is really easy to show that $P(2x+2)…
drfrankie
  • 509
2
votes
3 answers

Proof by induction using summation

I'm trying to figure out how to solve this equation by induction and I really don't know where to begin. I have seen some YouTube tutorials, but can't understand how I can go from $k(k+1)$ to $n+1$ in the equation. The task is: Use induction to…
Sirajuddin Asjad
  • 63
2
votes
2 answers

Prove by induction that $f(a,b) = \frac{a}{b} + \frac{b}{a} + \frac{1}{ab}$ is a multiple of 3 if it is an integer

I'm trying to solve the following problem by induction but I'm getting stuck. For positive integers $a$ and $b$, define $$f(a,b) = \frac{a}{b} + \frac{b}{a} + \frac{1}{ab}.$$ If $f(a,b)$ is an integer, prove that it is a multiple of 3. Proof by…
Jonathan
  • 514
2
votes
1 answer

What is the differences between proving by Deduction and Induction

I am just starting to learn about Mathematical proofs and so far I have learned about Mathematical Induction. I would like to know in its core, what is the main conceptual difference between proving something using Induction and Deduction. Example…
Matheus Minguini
  • 159
2
votes
4 answers

Prove by induction that $2^{4^n}+5$ is divisible by 21 .

After I did the first two steps in the induction , I am stuck in the last step ; to prove for $n+1$ that $2^{4^n}+5$ is divisible by $21$ , so I know that $2^{4^n}+5$ is divisible by $21$ is true . I want to prove for $n+1$ ( $n$ is natural…
Majd
  • 87
2
votes
3 answers

proof by induction that $a_n$ < 5, when $ a_{n+1} = \frac{6a_n + 5}{a_n + 2} $

$ a_1 = 1, $ $ a_{n+1} = \frac{6a_n + 5}{a_n + 2} $ (where n is a +ve integer) we have to prove by induction that $ a_n $ is always less than 5. so I used induction to prove that $ a_n $ is always positive. And i calculated $ a_2 = \frac{11}{3} $…
Vanessa
  • 23
2
votes
4 answers

Highschool induction (getting stuck on part 2)

I'm getting stuck on part 2 of an induction question. I genuinely wish to only receive a helpful hint for it and would rather be able to crack the problem myself. All help is highly appreciated! Here is the two parts of the question i) Prove by…
BooScout
  • 179
2
votes
3 answers

Prove geometric sum with induction

I'm not certain how to complete the proof: Question: Prove by induction that $1 − \frac{1}{3} + \frac{1}{9} − · · · + (\frac{−1}{3})^n = (\frac{3}{4})[1 − (\frac{−1}{3})^{n+1}]$, for every non negative integer $n$. Solution Base Step: Verify…
wulfgarpro
  • 123
2
votes
2 answers

Mathematical induction exercise

So I am having trouble with an exercise about mathematical induction. I have the following sentence: $1^{n+1}$ < $2^n$ for every n ≥ 3 Now, what I would personally do is: First prove that it is true for n = 3 $1^{3+1}$ = 1 < 8 = $2^3$ And assume…
Setin
  • 27
2
votes
1 answer

Proof by induction: String of characters

Here is a question I've been working on and so far, can't get anything. Suppose we have a string which is recursively defined as: base case: A = F recursive case: ( A N A ) such as a string X = ( ( F N ( F N F ) ) N ( ( ( F N F ) N F) N (F N F )…
slaughterize
  • 123
1 2 3
…
99 100