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. P

    Need help in finding some old papers on Induction effect

    Where and how can I find these two papers ? P. Debye, Physik. Z., 1920, 21, 178 H. Falckenhagen, Physik. Z., 1922, 23, 87 Thank you. Regards
  2. C

    Simple Proof by Induction Problem

    Hi, I devised this problem for myself as part of a bigger question that I'm working on, and am having trouble solving it. I think it involves a nested induction proof but I am not sure how to start. A tip on how to begin would be much appreciated. Thanks -Patrick Homework Statement S_0 =...
  3. R

    Magnetic Induction (Non-uniform Current)

    Homework Statement long wire of radius R0 carries a current density j given by Find the magnetic induction B inside and outside the wire. Homework Equations Current density: ##J=\frac{I}{A}## Ampere's law: ##\oint B.dl = \mu_0 I_{enc}## The Attempt at a Solution For the magnetic field...
  4. G

    Mathematical Induction where the base case starts above 1

    Homework Statement Find all natural numbers such that 2n ≥ (1+n)2, and prove your answer. 2. The attempt at a solution I can see this is true for n=0 and n>5. I try to prove this using induction as follows 20 =1≥ 1=(1+0)2 base case: 26 =64≥ 49=(1+6)2 so it is true for n=6 and suppose 2n...
  5. U

    Which case generates emf by electromagnetic induction

    1. a. coil is moved up and down in the same direction of flux. b. coil is rotated around an axis that is parallel to the flux. c. coil is rotated to the direction of the flux like a dynamo. 2. Electromagnetic inductions laws. 3. I'm really having a hard time...
  6. M

    Operating principle of a Six Phase induction motor

    Hi Sir, I have seen in many IEEE magazines that, a six phase induction motor is having two 3 phase windings spatially separated by an angle of 30 degrees electrical. I have attached the 6 phase induction motor diagram. But I am not able to understand how the motor operates i.e. how the rotor...
  7. I

    Charging a capacitor with induction

    Hello all, I am new here. So you see where my lack of understanding comes from; my background is as a Mechanical Engineer and I'm currently in my masters year. I haven't touched electronics since A levels but I was hoping somebody could point me in the right direction. These forums tend to...
  8. Seydlitz

    Ordinary Induction vs Complete Induction

    With my current understanding, the ordinary principle of Mathematical Induction is a method to prove, whether some statement ##P(x)## is true for all ##x## in ##\mathbb{Z}##, considering the truth of ##P(1)## and then the truth of ##P(k+1)## assuming ##P(k)## is true. For Complete Induction...
  9. D

    Use induction in a non standard way

    So I have to do an induction but I am not quite sure how to set it up. I have already proven that at each step I have either my intended result or I can advance one more step. I have also proven that there are a finite number of steps. Intuitively I have essentially completed the proof. I...
  10. M

    Real Analysis - Natural Number Induction

    Homework Statement Prove that if n is a natural number greater than 1, then n-1 is also a natural number. (Hint: Prove that the set {n | n = 1 or n in \mathbb{N} and n - 1 in \mathbb{N} } is inductive.)Homework Equations The Attempt at a Solution S(n) = \sum_{j = 2}^{n} j = 2 + 3 + \cdots...
  11. S

    MHB Proof of Inequality by Induction

    need help on this Show by induction that n^3 <= 3^n for all natural numbers n.
  12. T

    MHB Proving Induction: (1+1/n+1) to 2-1/n+1

    Show using induction that (1 + 1 / n + 1).(1 + 1 / n + 2). ... . (1 + 1 / n + n) = 2 - 1 / n + 1, n >= 1. I've tried everything with this question but the right hand side is not the same as the left hand side after substituting k+1 in the place of n, please help.
  13. B

    Is the charge induced on a conductor equal to or less than the inducing charge?

    electrostatic induction... when a conductor is charged by the method of induction,is its charge equal in magnitude to that of inducing charge or somewhat less?
  14. E

    MHB Is induction a circular way to define natural numbers?

    Sorry about the intriguing title; this is just a continuation of the discussion in https://driven2services.com/staging/mh/index.php?threads/5216/ from the Discrete Math forum. The original question there was how to introduce mathematical induction in a clear and convincing way. Since the current...
  15. sankalpmittal

    Problem regarding induction and Gauss' Theorem

    Homework Statement There are three concentric thin spherical shells A,B and C of radii a,b and c respectively. The shells A and C are given charges q and -q respectively and the shell B is earthed. Find the charges appearing on the surfaces of B and C. Homework Equations Gauss' Theorem and...
  16. matqkks

    Exploring Mathematical Induction: Impactful Examples and Real Life Applications

    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 introducing mathematical induction? I need something which will have an impact. Are there any real...
  17. matqkks

    Is Mathematical Induction the Key to Understanding Finite and Infinite Concepts?

    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 introducing mathematical induction? I need something which will have an impact. Are there any real...
  18. S

    Mathematical Induction on two Matrices

    Homework Statement (1 1)^n = (1 n) (0 1) (0 1) Prove this through mathematical induction. Homework EquationsThe Attempt at a Solution I've replaced n with 1, so I've done that far. Then I said k = n. Then replaced all n with (k+1). I'm really stuck...
  19. O

    Faraday's Law of Induction; Why is it the *enclosed* magnetic flux?

    Hello Reader, Basically in Faraday's law, a change of flux over time induces an EMF. I was wondering why is it the enclosed magnetic flux and NOT the magnetic flux through the conductor that we consider. More specifically, I understand that the flux through a rectangular surface would be...
  20. S

    Can charge induction occur in covalent solids?

    can covalent solids like sand,zeolite be induced with charge...
  21. S

    Charge Induction in Liquid: Localising Positive & Negative Charges

    can a liquid be induced with charge i.e. Can we create sites of localisation of positive and negative charges in a liquid?
  22. C

    Proving I_n \leq (2n/(2n+1))^n by induction for positive n

    I_n = \displaystyle \int_0^1 (1-x^2)^ndx, n \geq 0 Given that (2n + 1)I_n = 2nI_{n-1} proove by induction that I_n \leq \left (\dfrac{2n}{2n + 1} \right)^n for positive integers of n in the solutions, could someone explain how they got to step 1, and why we need to show step 2 to...
  23. H

    How to prove this question by induction

    Prove by induction that for all n≥ 1, dn/ dxn (e ^(x2) = Pn (x) e ^(x2) where Pn(x) is a polynomial in x of degree n with coefficient of x^n equal to 2^n I have problems trying to prove this question by mathematical induction. Please help...Really much appreciated
  24. P

    AP Physics E&M MC: Electric Field, Induction, LC Circuit

    Homework Statement Answer is B. Answer is E. Answer is E. Homework Equations E=F/q E=V/d Right hand rule for inductors The Attempt at a Solution Quite confused for these problems. For number 18, I'm quite baffled as to where the 0.04 meters even comes from. What I had...
  25. G

    Verify using Mathematical Induction

    Verify that 1(1!)+2(2!)+...+n(n!) = (n+1)! - 1 is true using induction This problem has me stumped...
  26. B

    Hamiltonian Path - Induction Proof

    Homework Statement Let G be a graph. 1. Let n be a natural number. Use induction to show for all n >= 2 Kn has a Hamiltonian path. 2. Explain how you could use the proof from #1 to show that for all n (natural number) n > 2 Kn has a Hamiltonian cycle. Homework Equations The...
  27. J

    Prove Induction: (1+2+4+...+2^n)+1=2^(n+1)

    To prove: (1 + 2 + 4 + . . . + 2^{n}) + 1 = 2^{n+1} , ∀n ≥ 0 . Basis Step n = 0: LEFT HAND SIDE: (2^{0})+ 1 = 1 + 1 = 2 RIGHT HAND SIDE 2^{0} + 1 = 2 Inductive Step: Assume (1 + 2 + 4 + . . . + 2^{k}) + 1 = 2^{k+1} Then (1 + 2 + 4 + . . . + 2^{k} + 2^{k+1}) + 1 = 2^{k+1} + 2^{k+1} +...
  28. D

    Can a Highly Charged Rod Strip All Electrons from Aluminum Foil?

    Homework Statement If there is a positively charged rod,which is placed near an aluminium foil,free electrons in the foil are pulled towards the rod.This makes the top end of the foil negative and leaves the bottom end short of electrons to become positive. I wonder if the rod has a very...
  29. S

    Binomial theorem induction proof

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  30. T

    Proof by Induction with Exponents

    Homework Statement By mathematical induction, prove that for n ≥ 1, 4/(7n - 3n). Homework Equations The Attempt at a Solution I got the base case down P(1): 7-3=4. Now the actual problem, 7n - 3n = 4x 7n+1 - 3n+1 = 7(7n) - 3(3n) =7(4x + 3n) - 3(7n - 4x) =21x+ (7(3n)) - (3(7n)) + 12x -This...
  31. N

    Fixed Speed Wind Induction Generator

    Hi, I'm sure there is a simple answer to the following question that's got me thinking lately but I'm unsure (it's more of a hypothetical question out of interest and not homework). If a simple older type fixed speed wind induction generator was connected directly to the grid via a gearbox...
  32. Y

    Prove the theorems using mathematical induction.

    Homework Statement Prove the theorems using mathematical induction. \forall n \in N, n\geq 4 \rightarrown2\leq n! Thanks in advance! Homework Equations The Attempt at a Solution First, check the base case which is n=4. \Rightarrown=4\geq4-True \Rightarrow42\leq4*3*2*1...
  33. M

    Another Proof by Induction Question

    Homework Statement Prove -(-x) = x. Homework Equations A2: x + y = y + x [additive commutativity] A5: x + (-x) = 0 M3: x(yz) = (xy)z [multiplicative associativity] M4: x (1) = x Lemma: (-1)(-1) = 1 Theorem c: (-1)x = -x The Attempt at a Solution -(-x) = (-1)[(-1)x] by...
  34. M

    Proof by Induction: 2n ≤ n! for All n ≥ 4

    Homework Statement Prove 2n ≤ n! for all n ≥ 4. Homework Equations (n+1)! = (n+1)n! The Attempt at a Solution First, notice P(4): 24 = 16 = 4*4 ≤ 4*6 = 4 * 3 * 2 * 1 = 4!. Supposing P(n) is true, check P(n+1): 2n+1 = 2*2n ≤ 2 * n! ≤ (n+1)*n! = (n+1)! Q.E.D...
  35. S

    Mathematical Induction: Find P(sub2)(A(subn)) & Prove (n*(n-1))/2

    Let A(subn) = {1,2,3,...,n} For any set B, let P(subk)B=the set of all subsets of B with exactly k elements. For example, P(sub2)({1,2,3})={{1,2},{1,3},{2,3}}. A) Find P(sub2)(A(sub1)), P(sub2)(A(sub2)), P(sub2)(A(sub4)), and P(sub2)(A(sub5)) B) Use mathematical induction to prove that the...
  36. G

    Mathematical induction question

    Hi all, I am revising on Proof by mathematical induction and I have came across a question I haven't found a way to work it out. \sum_{r=1}^n r(r!) = (n + 1)! -1 I understand the steps of proving by mathematical induction question. The ! is causing the confusion.
  37. G

    How Does Series Notation Equal the Highlighted Part in Induction Proof?

    Hi all, I am so sorry I don't know whether to post this. I am revising Proof by mathematical induction Further maths using an Edexcel FP1 book. The following is from chapter 6 example 3. I know how to work Proof by mathematical induction it is part of the example that I don't understand. I...
  38. C

    Math Induction: Where Does the >2xk Come From?

    Please refer to the image attached. Where does the > 2 x k come from? Based on the proposition, shouldn't it be > k+1?
  39. B

    Magnetic Induction Through a Moving Wire

    Homework Statement A very long u-shape conducting wire is placed vertically in a constant magnetic field. The magnetic field is perpendicular to the plane of the wire. The strength of the magnetic field is B = 0.9 tesla. A conducting rod is placed on the wire, as shown in the picture. The...
  40. V

    Base Case in Strong Induction Proofs

    I am dealing with sets of problems that go as such: "How many n-cent postages can be formed from x and y cent stamps" For instance, I am doing a problem where x and y are 4 and 11 respectively. I don't understand how to determine a base case. I know that I must proof P(k + 1) for all P(i)...
  41. MarkFL

    MHB Tom's question at Yahoo Answers regarding proof by induction

    Here is the question: Here is a link to the question: Proove the following by mathematical induction? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  42. E

    Torque due to electromagnetic induction

    Homework Statement A circular coil of radius r carries a current I. A magnetic induction B acts at right angles to a diameter of the coil. Show that the current experiences a torque T about the diameter given by T=Iπr^{2}Bsinω, where ω is the angle between the normal to the plane of the coil...
  43. M

    Voltage Increase through Induction

    I am new to the field and have stumbled across an issue. The current project I am working on is requiring me to calculate how many windings I need to increase voltage in a system from 12v to ~400v. The Initial winding will be fed from a 0 A.W.G. braided copper wire and the outgoing feed will...
  44. J

    Mastering Mathematical Induction: Solving for the nth Term

    Homework Statement 1-1/2+1/4-1/8+...nth term= [(2^n-(-1)^n)]/3(2^(n-1))] I just can't get the nth term. I realize that the denominator should be 2^(n-1) and that, because the sign constantly changes there needs to be something like -1^n so I came up with the equation -1^n/-2^(n-1) but this...
  45. M

    Impossible to have solution to the equation. Induction Proof Problem

    Homework Statement For all integers n, it is impossible to have a solution to the equation 4^n = a^2 + b^2 + c^2 where a, b and c are all positive integers. (Hint: Notice that 4^n = 2^2n is a perfect square. Show (prove) that if m^2 = a^2 + b^2 + c^2, then we must have that a, b and c...
  46. J

    Engineering P= 1.732 X 230V X 42A X 0.85 = 10,731.06 W or 10.73 kW

    3 PHASE INDUCTION MOTOR A 15 hp/230V, three phase induction motor has a full load current of 42A per terminal. 15HP= 11kW find: a) Full load kVA input b) If the full load power factor is lagging 0.85, what is the kW input. My question: How do I know whether its connected Wye or Delta?a) kVA=...
  47. E

    Volts/hertz control of three phase induction motor.

    I am doing project on open loop v/f control of three phase induction motor. How to decide the frequency and amplitude of triangular wave(carrier wave) for sinusoidal PWM ?
  48. A

    Exploring the Logic of Mathematical Induction

    Hello I'm learning about proofs and in my book there's a sect. On mathematical induction. And I'm trying understand why this makes it true for all values. 1+3+5...2n-1=n^2 Suppose that the formula is known to be true for n=1, and suppose that as a result of assuming that it is true for n=k...
  49. D

    Question about grounding and induction charging

    Hello, I have a question that I hope someone here can help me with. In all of the descriptions about induction charging and grounding, the incoming negative charge is depicted as being applied to one side of a neutral, uncharged object. If I understand correctly, according to the law of...
  50. G

    Enquiry on flexible-elastic induction heating

    Hi, I am a designer currently working on alternative induction cooking methods. The question is, whether it is possible to have a flexible/elastic cooking surface (something like a mat, that would roll up, fold up, expand etc.) A project similar to this, is here...
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