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. C

    Electromagnetic Induction and a coil of wire

    The magnetic field that is oriented perpendicularly through a 9.2 cm diameter coil of wire drops from 6.4 T to 6.1 T in 0.076 seconds. What is the emf induced in the coil? Trying to answer this. My question is, if they are perpendicular then does that mean that cos90 makes phi = BAcos theta...
  2. evinda

    MHB Proving $T(n)=\Theta(n \log n)$ by Induction

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  3. S

    Generator / induction formulas help

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  4. B

    Induction Generator connected to a standalone Synchronous generator

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  5. J

    Induction Heater - HalfBridge Series Resonance Circuit

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  6. topsquark

    MHB Induction Proofs & Negative Proofs: Examining POTW 135

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  7. P

    How Does Induced EMF Vary Inside a Solenoid?

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  8. H

    Induction motor generator capacitor

    Hope I can get some help here. I have a 1-1/2 hp. 115/230 volt single phase induction motor I want to use as a small generator. currently it's wired for 115 volt I need help with the math to calculate the proper capacitor/s to use. Running no load amps as a motor draws 18 amps. Name plate...
  9. T

    Composite wire section as it relates to induction

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  10. S

    Integral problem / mathematical induction

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  11. Thomas Rainey

    How to Approach an Induction Proof for a Product Ratio Inequality?

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  12. evinda

    MHB Proving by Induction: Solving Recurrence Relation

    Hello! (Wave) I want to prove by induction, that the solution of the recurrence relation $T(n)=2T \left ( \frac{n}{2} \right )+n^2, n>1 \text{ and } T(1)=1$ is $n(2n-1)$. We have to suppose that $n=2^k, k \geq 0$, right? Do I have to prove the solution by induction with respect to $n$ or to...
  13. PcumP_Ravenclaw

    Simple Proof for using Induction

    Dear All, I am trying to understand this proof for using induction. Please help me! As per the book "Alan F beardon, Abstract algebra and geometry" The following... Quote: Proof: Let B be the set of positive integers that are not in A. Suppose that B = ∅; then, by the Well-Ordering Principle...
  14. W

    Induction Foil Iron vs. Aluminum

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  15. S

    Odd phenomena with an induction cooktop

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  16. W

    Proof by Induction: Solving Algebraic Exercises

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  17. J

    How do I prove the statement p(n)= n >(or equal to) 0 using induction?

    Homework Statement a)Prove, by induction on n, that for all n ∈ N(natural numbers), Distance(x,y)<or equal to 0b)Prove, by induction on n, that for all n ∈ N(natural numbers), Distance(x,y)<or equal to n Homework Equations proof by induction: Base case P(1) then assume p(k) is true then...
  18. F

    Proving Orthogonality by Induction in Second Quantization for Bosonic Systems

    Hello, I'm currently studying second quantization. I need to prove <n^\prime| n>=\delta_{n^\prime n} by mathematical induction in the number of particles representation. However I don't know how to do this proof having two natural numbers n and n^\prime. Must I prove it holds for <0|0>, <0|1>...
  19. B

    Interpretation of induction in terms of symmetry transformations

    It just occurred to me that induction can be seen as a statement quite analogous to that of "a function whose derivative is 0 on an interval is constant on that interval". Suppose there is a property P about the natural numbers that we want to prove. Then let P: N -> {0, 1} be a function for...
  20. M

    How Do Faraday's Law and the EMF = BLv Equation Relate?

    Alright, I need to put my question in the context of the probably ubiquitous example of a conductor rod moving perpendicularly over a magnetic field to generate an emf between the two sides of the rod. That emf is equal to BLv (magnetic field, length, velocity) according to my book(studying for...
  21. C

    Expansion of series and possible induction

    Homework Statement Consider the infinite series $$\frac{x}{e^x - 1} = A_o + A_1 x + \frac{A_2}{2!}x^2 + ... + \frac{A_n}{n!}x^n + ...$$ Determine that ##1 = A_o,\,\,\,\,\,0 = A_o/2! + A_1,\,\,\,\,\,0 = A_o/3! + A_1/2! + A_2/2!##. Show that for ##n > 1##, one can write the relations as $$(A+1)^n...
  22. johann1301

    Is this induction proof correct?

    Homework Statement The sequence {xn} is given by the recurrence relation xn = cos(xn-1)sin(xn-2) for n ≥ 2 and x0=2 and x1=1,4. Show by induction that 0 ≤ xn ≤ 1 for all integers n ≥ 2. The Attempt at a Solution We formulate a statement: Pn: 0 ≤ xn = cos(xn-1)sin(xn-2) ≤ 1...
  23. S

    Electrostatic Induction: Conductor vs. Dielectric Response Time

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  24. nuuskur

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  25. R

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  26. K

    Proof by mathematical induction

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  27. Shackleford

    Grad School Struggles: Finding a Formula and Proving it with Induction

    I started graduate school this week after getting my BS in May of 2012. I'm having to review a lot of material. Heh. I've finished most of the homework, but I'm a bit stumped on two easy problems. The induction proof is no problem; I just haven't been able to find a formula. The nth partial...
  28. mxam

    Is n^n+1 a Valid Formula for the Sum of n^n Series?

    Can you help me with this exercise? 1^{1}+2^{2}+3^{3}+4^{4}+...+n^{n} = n^{n+1} Thanks! PD. I was trying to solve, and i have this: 1^{1}=1^{1+1} = 1 = 1 a) k^{k+1} b) k+1^{k+1} = k+1^{(k+1)+1} a in b) k^{k+1} + k+1^{k+1} = k+1^{(k+1)+1}...
  29. A

    A question about electromagnetic induction

    Hi guys I just want to know why changing magnetic flux in a coil induces emf? Thanks in advance.
  30. johann1301

    Proving Induction: n∑(1/√n) > 2(√(n+1) -1)

    Homework Statement Prove this by induction n ∑(1/√n) > 2(√(n+1) -1) i=1 The Attempt at a Solution I have tried for several hours, with no real progress. When i assume that it works for n=k, and then try to prove for n=k+1 i get 2(√(k+2)-1)... i don't know how to partition this...
  31. M

    Electromagnetic induction confusion

    I was just thinking about the "changing magnetic field through a loop induces an EMF" and thought of a conceptual question I'm having trouble with. So, imagine you have an open surface where there's a changing magnetic flux that you know (say its a plane of magnetic field coming toward you...
  32. P

    Magnet configuration for spinning permanent magnet induction heating

    Watch this video: Permanent magnets moving relative to a stationary copper tube generate Eddy currents which result in Joule heating of the copper. Simple enough. Let's change the problem to a rotor with 4 (circumferential) permanent magnets rotating inside a stationary (aluminum or...
  33. gfd43tg

    Mathematical induction ''all horses are the same color''

    Homework Statement Find the error in the following proof that \all horses are the same color" 4. Claim: In any set of h horses, all horses are the same color. Proof: By induction on ##h##: Base Case: For ##h = 1##. In any set containing just one horse, all horses are clearly the same color...
  34. E

    Electrostatic induction - rod and sphere

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  35. gfd43tg

    Proving Relationships with Mathematical Induction

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  36. I

    Is Mathematical Induction Proven Correctly for $S_{k+1}:2^{k+1}>(k+1)^2$?

    $S_k>k^2$ $S_{k+1}:2^{k+1}>(k+1)^2$ $2*2^{k+1}>2(k+1)^2$ $2^{k+2}>2(k+1)^2$ Assume $x=k+1$ $\frac{2^{x+1}}{2}>x^2$ $2^{x+1}*2^{-1}>x^2$ $2^x>x^2$ right? $2^{k+1}>(k+1)^2$
  37. I

    Using Mathematical Induction to Prove a Summation Formula

    $S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k=6(6^k-1)$$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k+ 5\cdot 6^{k+1}=6(6^k-1)+5\cdot 6^{k+1}$ what do i do now? to prove $S_{k+1}$
  38. Greg Bernhardt

    What is mathematical induction

    Definition/Summary Mathematical Induction is a method of proving a series of mathematical statement labelled by natural numbers. This method usually involves two steps. First one proves the base case, then one shows that if the statement holds for some natural number, it holds for the...
  39. I

    Is $k \geq 3$ enough to conclude $(k - 1)^2 > 2$ in mathematical induction?

    Did i do this right? $2^n>n^2$, $n \ge 5$ $S_{k+1}: 2^{k+1}>k^2+2k+1$ $2^k>k^2$ $2(2^k)>2k^2$ $2^{k+1}>k^2+k^2$ And $k^2+k^2>k^2+2k+1$ RHS: $k^2+2k+1$ So $2^{k+1}>(k+1)^2$
  40. I

    How Does Mathematical Induction Work?

    I don't know what forum to post this under. PLEASE HELP ME THOUGH! Principle of Mathematical induction: Let $S_n$ be a statement concerning the positive integer n. Suppose that, $S_n$ is true. For any positive integer k, k $\le$ n, if $S_k$ is true, then $S_{k+1}$ is also true. Then $S_n$ is...
  41. W

    Time Constant Induction question

    Hi all, I need your help with a Time constant question? I am studying for an exam via distance learning and don't feel they have covered this question, question taken from past paper. 2. A coil of inductance 2 H and unknown resistance is connected to a D.C. supply of 100 volts. After 4 ms the...
  42. I

    Derivation of Faraday's Law of Induction?

    I was wondering if someone could show me the derivation of Faraday's Law of induction from the more fundamental Lorentz Force.
  43. M

    Single phase Induction motor Design

    I want to design a single phase induction motor (1/3 HP,50Hz,220V). I face a problem to design the main dimension of stator and Slot shape and Rotor skew angle . Would you give some idea for design ? or any related book and software ?
  44. H

    How Do I Simplify the Right Side of the Mathematical Induction Equation?

    Original Equation: 5+6+7+...+(n+4)=1/2n(n+9) Ok, I've tried everything to understand this. I'm just not getting it. I understand everything (n=1, k+1, etc) up until this point: "To continue with proof what must be done?". I know you must simplify the right side, but I don't understand how they...
  45. 9

    Koch Snowflake Proof by Induction.

    Hi, I was wondering if there is a way to prove the area of the Koch Snowflake via induction? At the moment I have the equations: An+1=An+\frac{3√3}{16}(\frac{4}{9})n and An=\frac{2√3}{5}-\frac{3√3}{20}(\frac{4}{9})n These two don't seem to work together very well when trying to prove by...
  46. gfd43tg

    Mathematical Induction: Proving Inequalities with k and k+1

    Homework Statement Homework Equations The Attempt at a Solution Hello, I am working on the problem in the attached image regarding induction based on the inequality 2^n \geq n + 1 I am confused how to do this by proving that if it is assumed to be true for ## n = k ##, then how it is true...
  47. J

    Proving the Induction Step: \sum_{j=1}^{k+2} j \cdot 2^j = (k+2)\cdot 2^{k+4}+2

    Homework Statement prove by induction \sum_{j=1}^{n+1} j \cdot 2^j = n \cdot 2^{n+2}+2; n \ge 02. The attempt at a solution P(0) \sum_{j=1}^{0+1} j \cdot 2^j = 0 \cdot 2^{0+2}+2 2+2 here is where I need some help is P(k) \sum_{j=1}^{k+1} j \cdot 2^j = (k+1) \cdot 2^{k+3}+2 ?? then...
  48. J

    Finding the sum of 1^3 + 2^3 + + n^3 by induction

    1^3+2^3+...+n^3 = \left[ \frac{n(n+1)}{2}\right]^2; n\ge 1 P(1) = 1^3 = \frac{8}{8} = 1 P(k) = 1^3+...+k^3 = \left[ \frac{k(k+1)}{2}\right]^2 (induction hypothesis) P(k+1) = 1^3+...+k^3+(k+1)^3 = \left[\frac{(k+1)(k+2)}{2}\right]^2 I start getting stuck here I foiled it out then let m =...
  49. C

    Three phase Induction Motor torque/slip HELP NEEDED

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  50. H

    Induction motor and sequence reactance

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