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. sushichan

    Is (n^2+3)(n^2+15) divisible by 32 for odd positive integers n?

    Homework Statement Prove that (n2+3)(n2+15) is divisible by 32 for all odd positive integers n. Homework Equations I suppose we are supposed to use mathematical induction since it is in that chapter, but the following questions specifically state that we should use induction but this question...
  2. A

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  3. B

    A.c. current and induced force

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  4. A

    Is Induction Proof Needed for this Limit Problem?

    Prove that for fixed s > 0, we have lim e^-sN ( s sin bN + b cos bN ) = 0 N -> This problem was in a chapter on Laplace Transforms. I'm assuming this will require proof by induction.
  5. J

    Electromagnetic induction question

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  6. R

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  7. ranju

    Finding Full Load Current & Torque in Induction Motors

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  8. N

    What is the working principle of an isolated induction generator?

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  9. N

    Poly-phase Induction generator

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  10. J

    Magnetic field and electric field induce one another forever

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  11. J

    Electromagnetic induction of a closed circular wire

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  12. C

    Mathematical induction example

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  13. D

    Homework Help on Kirchoff's Laws and Magnetic Induction

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  14. N

    Why crawling happens in induction motor?

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  15. S

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  16. moriheru

    Proving ∑mk=1k2=1/6(323+3m2+m) by Induction

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  17. Pami

    Prove gamma (n+1/2) = (2npi^1/2)/(n4^n) by induction

    I tried solving this question this way: Gamma(n+1/2) =(n+1/2-1)gamma(n+1/2-1) =(n-1/2)gamma(n-1/2) =(2n-1)/2 gamma (2 n-1)/2 Don't know what to do next
  18. A

    Electromagnetic induction of two wires

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  19. 1

    Proof by induction: n^3 < n for n >=6

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  20. Jackson Lee

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  21. ellipsis

    Prove recurrence relation via mathematical induction

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  22. T

    Strong Induction Proof with a Floor

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  23. Z

    Elastic Potential Energy to Aid Electromagnetic Induction

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  24. evinda

    MHB Proving $T(2^k)=2^k \cdot k$ with Induction

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  25. Colleen G

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  26. D

    MHB Proof by Induction: Solving 34n-1 Divisible by 80

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  27. dirtybiscuit

    Proof by Induction: Prove Fn-1Fn+1 - Fn = (-1)^n

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  28. ElijahRockers

    Fourier series coefficients: proof by induction

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  29. A

    Electrostatic induction in different size spheres

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  30. M

    Induction cooking & heat distribution

    I've been trying out induction cooking on a small one-burner unit, with a large cast iron pan. I'm getting the expected hot spot in the center of the pan, and a serious temperature falloff toward the sides. No surprise there - the induction element is about 7" (I dismantled the unit to check)...
  31. N

    Proof by Mathematical Induction

    I am confused on a math problem. I am supposed to use mathematical induction to show that (Summation with n on top and i=1 on the bottom) (2i-1) = 1+3+5+...+(2n-1)= n^2
  32. P

    3 Phase Induction Motor Full Load Power Factor

    Hi, If the voltage, full load current and full load output power (kW) of a 3 phase induction motor are known, and the efficiency is assumed to be 90% (is this realistic), am I correct in thinking that the full load power factor can be approximated (due to assumed and not actual efficiency) as...
  33. NATURE.M

    Can Sequence r_n Be Bounded by O(log2(log2 n))?

    Homework Statement We are given the sequence r defined by: r1 = 1, and rn = 1 + rfloor(√n) , n≥2 We need to show by induction that rn is O (log2 (log2 n)). The Attempt at a Solution Definition of big oh: ∃c∈ℝ+, ∃B∈ℕ, ∀n∈ℕ, n≥B => f(n) ≤ cg(n) [/B] So the basic proof format is fairly simple...
  34. A

    Three phase induction motor efficiency

    Hi, Is there a difference in efficiency between a three phase motor and a single phase motor, in terms of power in against power out. If so, could you please explain why one is more efficient than the other. The reason I ask is because I'm researching the development of the induction motor for...
  35. L

    Does an Induction Cooking Pan Need to be Completely Ferrous?

    I ask, "Does a ferrous pan for Induction Cooking need to be Completely Ferrous?" In other words, could just the bottom of the pan be a ferrous material for good inductive heat induction? Could I have a ferrous bottom, and ferrous sides that turn to a non-ferrous material after a couple...
  36. carllacan

    Where is the EMF located in a circuit under induction?

    Suppose we have a closed circuit with a time-varying magnetic flux. An EMF is inducted on the circuit. Where would we put the battery representing the EMF? For example, suppose we have a resistance in the circuit and we want to know the intensity through it. Would we just set the voltage...
  37. evinda

    MHB Prove by induction that the function is injective

    Hi! (Wave) The set $\mathbb{R}$ of real numbers is not countable. Proof: We define the function $F: \{0,1\}^{\omega} \to \mathbb{R}$ with the formula: $$(a_n)_{n \in \omega} \in \{0,1\}^{\omega} \mapsto F((a_n)_{n \in \omega})=\sum_{n=0}^{\infty} \frac{2a_n}{3^{n+1}}$$ Show that $F$ is 1-1...
  38. T

    Induction and flux linkage clarification

    So if you have a single inductor then I assume that there is no flux linkage and that L = flux*N / i = L11 and so that flux is total leakage flux. If you have two inductors then the flux leakage is any flux that isn't flux linkage (the self flux is flux leakage). (flux21 is the flux in 2 from...
  39. 2

    Magnetic fields in transformers and back emf?

    i am having some trouble with understanding the magnetic fields inside transformers and how they relate to back emf. In particular, I am considering the two cases of current passing through the primary coil but not the secondary coil, and then with the secondary coil as part of a complete...
  40. H

    Can Induction Prove 3^n ≥ n2^n for All n ≥ 0?

    Homework Statement The question asks me to prove inductively that 3n ≥ n2n for all n ≥ 0. Homework EquationsThe Attempt at a Solution I believe the base case is when n = 0, in which case this is true. However, I cannot for the life of me prove n = k+1 when n=k is true. I start with: 3^k ≥...
  41. B

    Induction Machines: Torque Speed Characteristics & Benefits

    < Mentor Note -- thread moved to HH from the technical engineering forums, so no HH Template is shown > hello really need some help i would like to know how would the torque speed characteristic for a "constant frequency variable voltage supply" and 'variable frequency variable voltage...
  42. J

    Is an induced emf always produced by a change in flux?

    Faradays law tells us that a change in flux induces an emf. Now consider the phenomenon of motional emf. It is observed across the ends of an open conductor (ie one which is not in a circuit). It is always discussed in connection to faradays law. Where is the change in flux in the case of a...
  43. M

    Inductive transfer via mag. fields, max power transfer?

    Hey guys; I'm looking for info on the maximum power transfer using the magnetic induction method. The current through a coil, then catch the mag field with another coil idea. Mainly looking for high energy transfer methods, maybe some equations that describe it. A friend and i are playing...
  44. Hijaz Aslam

    Charge induced on capacitor plates

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  45. A

    How does Faraday's law of induction relate to Special Relativity Theory?

    I need to what is induction as it is so confusing. Thanks for the help.
  46. B

    Calculating Induced Voltage in a Flying Airplane through Earth's Magnetic Field

    Homework Statement Assuming you know general specifications of the airplane such as span of the wings and maximum speed, calculate the voltage that is induced when the airplane is flying horizontally through Earth's magnetic field where both the horizontal component and vertical component of...
  47. S

    What is the effect of the capacitor value on an induction generator?

    Can anyone tell me what effect the value of the capacitor in an induction generator has. I am generating 230V, at an unknown speed. I am not particularly concerned about the generated frequency, but I want to run the thing at higher revs, and would like to know whether I need to reduce the...
  48. M

    Proof by Induction: Sum of 2^n - 1 Explained Step by Step

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  49. W

    Squirrel cage induction motors - rotor issues

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  50. Faradae

    Hysteresis in Induction Cooking

    Hi all, I was researching into Induction Cooking (using induction plates) recently and chanced upon a few websites that mentioned about hysteresis in Induction Cooking. However I don't really get how does hysteresis have a part to play in Induction Cooking. :/
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