Induction Definition and 999 Threads

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.

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  1. I

    Mathematical induction with a summation in the problem.

    Hi, I'm currently taking Discrete Mathematics and I'm working on a mathematical induction problem that's a little different than usual because it has a summation in it. What I basically want to know is did I do parts A and C correctly? Homework Statement Homework Equations The Attempt at a...
  2. T

    How Many Ways to Place Divisors Around a Circle in Mathematical Induction?

    I want to now the answer of this question and I think it relates to mathematical induction. The question is: -Suppose is a natural number. In how many ways can we place numbers around a circle such that each number is a divisor of the sum of it's two adjacent numbers? Who can answer this...
  3. H

    Transfinite Induction: Understanding & Examples

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  4. D

    Mathematical induction problem

    Hello! First of all I have like 5 exercises I don't quite understand so will it be a problem if I create 5 new topics in the next 24h? Homework Statement Prove, by using mathematical induction that if x+1 \geq 0 then (1+x)^n \geq 1+nx. Homework Equations The Attempt at a Solution Basic step...
  5. N

    Combinatorics - Mathematical Induction?

    Hello, I am having trouble solving this problem. Maybe I'm just overreacting to it. In my two semesters in discrete math/combinatorics, I've never seen a problem like this (with two summations) and been asked to prove it. Can some one help? \sum^{n}_{i=1} i^3 = \frac{n^2(n+1)^2}{4} =...
  6. Saitama

    Charge Induction: Understanding Electric Field Effects

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  7. D

    Faraday's Law: Magnet Motion & Energy Exchanges

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  8. E

    Electromagnetic induction question

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  9. I

    Overheating of Three phase Induction motor

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  10. C

    Bit of confusion with coils and electromagnetic induction

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  11. D

    Mathematical induction problem

    Homework Statement Hello! This is I want to prove using Mathematical Induction: 1+3+5+...+(2n-1)=n^2. The problem is: I don`t understand very much about Mathematical Induction :(Homework Equations The Attempt at a Solution Suppose n=1. Then 1=1. Now suppose 1+3+5+...+(2n)=(n+1)^2. Then...
  12. throneoo

    Need help with an electromagnetic induction problem concerning two rods

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  13. M

    Proof by induction, puzzles by answer

    Ok, so there's this proof by induction question. I looked at the answer for it but I don't understand it. How does it get from being 2^(k+1) to being 2^(k+2)?
  14. A

    Proof by induction of the sum of 2 squares

    Homework Statement (a^2+b^2)(c^2+d^2)=(ac-bd)^2 +(ad+bc)^2 prove by induction that r=a1,a2,a3...an where the a's represent the sums of 2 squares. it itself is a sum of two squares. Check it with: 2=1^2+1^2, 5=1^2+2^2, 8=2^2+2^2, for r=160, r=1600, r=1300, r=625 Homework Equations...
  15. Artlav

    Induction heating of non-magnetic metal?

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  16. R

    Formulas for designing an induction generator

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  17. T

    Induction on the n-dimensional, radially symmetric wave equation

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  18. A

    Proving Induction from the Well-Ordering Principle

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  19. B

    Electromagnetic Induction Without Air Core

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  20. I

    Induction by alternating voltage?

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  21. A

    Electromagnetic induction question

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  22. S

    Charging a widget through induction

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  23. H

    Proof by Induction Homework: Proving F_n|F_{kn}

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  24. D

    Proving inequality by mathematical induction

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  25. M

    Question about magnetic induction

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  26. P

    Can Mathematical Induction Prove the Existence of a Fourth Dimension?

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  27. S

    Induction and alternating current

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  28. T

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  29. F

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  30. C

    Electromagnetic induction inside a field with a contraint

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  31. B

    Fibonacci Sequence Induction Problem

    Homework Statement Show that for all n greater than 1: fn = \frac{1}{\sqrt{5}}{(\frac{1+\sqrt{5}}{2})n - (\frac{1-\sqrt{5}}{2})n} Homework Equations f1 = f2= 1 fn+2 = fn+1 + fn The Attempt at a Solution I'm pretty sure it's by induction, but I'm not sure how to start.
  32. X

    Proof by induction an expression

    I'm trying to prove by induction the expression: \sum_{i=1}^{n-1}(n-i)=\frac{n(n-1)}{2} For the base case, n=2, S(2)=\frac{2(2-1)}{2}=1 For S(n+1)=\frac{(n+1)((n+1)-1)}{2} I have: S(n+1) = \frac{n(n-1)}{2} + (n+1) <--- Is this correct? I don't know what is the term for n+1. Any help?
  33. N

    Hi all. Induction heater question

    Hi, new to the forum. Great place but much of the info far past my interlectual understanding. :frown: I have been looking at the idea of an induction heater for use in my job. I'm a mechanic by trade, nicknamed proffessor screwtop by my workmates :) Snap recently introcuced the...
  34. C

    Mathematical Induction problem.

    Homework Statement (1/2!)+(2/3!)+(3/4!)+...+(n/(n+1)!) a) calculate for a few small values of n. b) Make a conjecture about a formula for this expression c)Prove your conjecture by mathematical induction. Homework Equations The Attempt at a Solution So for the first part I just used values...
  35. Coffee_

    Solving the Paradox of Electromagnetic Induction

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  36. C

    Engineering Graphing Induction motor characteristics

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  37. C

    RPM vs Torque, I, Mech HP, Eff, and Fp of an Induction Motor

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  38. R

    How Do You Prove Matrix Powers Using Mathematical Induction?

    Hey guys I am in precalculus right now and we just started picking up mathematical induction. Our teacher assigned us a problem that I am stumped over and I tried looking all over for a clear explanation online but I can't find anything remotely helpful. The question is: Use mathematical...
  39. S

    Proof of Closed Set by Induction

    Hello all, I am trying to prove that a set is closed by induction. Specifically, let me define Let B_t be sets, and A_T:=sum{B_t: t=1, .., T}=Sum{b_t: b_t in B_t, and t=1, ..., T} A property that these sets have is that B_s is a subset of B_t for s<=t. I try to prove A_T is closed by...
  40. T

    Explanation of Electromagnetic Induction

    I took a physics competition recently and found I missed 6 questions on electromagnetic induction and the like. I've always been shaky here, so I'm wondering if anyone can give me an online resource, or just explain to me exactly how it works (i.e., Lenz's law, etc., etc.)? I'm sorry for not...
  41. M

    Induction Method: My Question About Theorem in Finite Groups

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  42. M

    Induction Method: Applying to Any Group or Subgroup?

    My question is about the induction method. This was in a theorem that I read. Let H be a normal subgroup of a finite group G. If G satisfies H-some statement then G is solvable. In the poof I have this. Let G be a counter example of minimal order. Let N be a proper normal subgroup of G. Since...
  43. M

    Electromagnetic Induction of a permanent megnet

    Hello everyone, How are you doing? I have a doubt about electromagnetic induction, in three particular cases. I need to confirm that I have the right concepts, so I ask for your help. The main problem: Imagine that you have a permanent magnet, axially polarized and rotating on its...
  44. R

    Proving a[n]≤2^n using Mathematical Induction

    1. Define a sequence of numbers in the following way: a[k]=a[k-1]+a[k-2]+a[k-3] for k≥3, s.t. a[0]=1, a[1]=2, a[2]=3, a[3]=6... *The numbers in brackets will be subscripts for the whole problem Prove that a[n]≤2^n using complete mathematical induction. This is what I have so far. We'll use...
  45. C

    Magnetic field induction enigma?

    Hello all! I was researching about magnetism to investigate the processes which occur when a magnetic object is stuck onto a magnet. I gather that this will be due to the alignment of magnetic domains and as such, the sum of all these electron spins in one direction will cause the attraction to...
  46. H

    Proof (x^n)'=nx^n-1 by complete induction

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  47. W

    How Can the Fibonacci Sequence Be Proved by Induction?

    I've been having a lot of trouble with this proof lately: Prove that, F_{1}F_{2}+F_{2}F_{3}+...+F_{2n}F_{2n+1}=F^{2}_{2n+1}-1 Where the subscript denotes which Fibonacci number it is. I'm not sure how to prove this by straight induction so what I did was first prove that...
  48. A

    Metal Spheres; Interaction between conductors; induction

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  49. J

    Why Is EMF Induced Only in One Direction in a Moving Conductive Square?

    Homework Statement If I move a SQUARE conductive layer in a forward direction, with a magnetic field pointing vertically upward why is an EMF only induced in the left-right plane NOT the forward-backward plane??Homework Equations The Attempt at a Solution I would have thought that as BOTH...
  50. B

    Law of Induction: What is the value of B in the equation ε = (BA)/t?

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