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. Ackbach

    MHB Electromagnetic Induction: Problem 30.85 in Halliday and Resnick and Walker

    $\newcommand{\uvec}[1]{\boldsymbol{\hat{\textbf{#1}}}}$ There is a uniform magnetic field $\mathbf{B}$ directed along the axis, and confined to a cylindrical volume of radius $R$. The magnitude of $\mathbf{B}$ is decreasing at a constant rate of $10$ mT/s. In unit-vector notation, what is the...
  2. M

    Induction Cooking: Eddy Current vs Hysteresis Frequency

    In induction cooking which effect is more dominant.EDDY CURRENT OR HYSTERESIS and at which range of frequencies?
  3. D

    MHB Solving Math Induction: Step-by-Step Guide

    Could anyone please help with this question regarding mathematical induction; step by step procedure/ solution would be greatly appreciated. Thanks!
  4. S

    Engineering How Do You Calculate Torque and Efficiency in a Three Phase Induction Motor?

    Homework Statement A three phase cage rotor induction motor running at full load draws a stator current of 60A at a power factor of 0.8 lagging from a 415v, 50Hz Supply. Under the conditions the stator loss is 3.6kW, the rotor loss is 1.3kw and percentage slip is 4.5%. If the shaft power...
  5. J

    Is Induction the Key to Proving n^2 > n+1 for n ≥ 2?

    1. Prove that n2 > n+1 for all n≥2 2. First, (2)^2>2+1...4>3 Now for the induction step, (n+1)2 > 2n+2 Can I just show that n2+2n+1 > 2n+1+1 because 2n+1 = 2n+1 for both sides and n2>1 for all n>1 I'm pretty sure what I did is not acceptable fro this to be considered proved
  6. G

    Does decreasing input voltage affect the slip in an induction motor?

    Hello all My question is: Induction motor with nominal load rotates with nominal speed. How the slip will be changed if we decrease input voltage in 10%? I think that if we decrease the voltage, than the current will be decreased also. Than, the moment will be decreased and mechanical...
  7. C

    Exercise of Electromagnetic Induction

    Homework Statement Homework Equations That is my solution: The Attempt at a Solution i solve my problem without ø=30°, I confused what this angle was used to do?
  8. R

    MHB Proof Using Induction: Discrete Maths Problem Solving

    Can someone help me solve the following problems from Discrete Maths, using induction to prove them Thanks in advance
  9. C

    Induction Charge: Losing Electrons? Help!

    I have a confusing, i think it lose electron instead of gaining electron. Help me, please
  10. F

    Calculating Power Dissipation in a Thin Conducting Disc with Induced Currents

    Homework Statement Thin conducting disc, radius a, thickness b and resistivity p (assumed to be large enough induced currents produce no magnetic field). There is a uniform B field B0sin(wt) parallel to its axis. I first had to find the electric field a distance r from the disc axis in the...
  11. F

    Solve Induction Slider Homework: Find AB Voltage & Current

    Homework Statement An induction device is set up as shown below. It has a U-shaped conductor of negligible resistance along which a conducting slider of resistance R moves in the x direction. The assembly is immersed in a uniform magnetic flux density B of magnitude B0 that is directed out of...
  12. K

    Proving |a-b| ≤ $\sqrt{a^2 + b^2}$ Using Induction

    Homework Statement Prove for any ##a,b \in \mathbb{R^+} : |a-b| \leq \sqrt{a^2 +b^2} ## The Attempt at a Solution How should I start? Can I use induction? Should I use contrapositive?
  13. K

    MHB Proving Positive WFFs with Induction

    hello, i was wondering if anyone could help me with this induction proof. thank you A positive well-formed formula in the Propositional Calculus is a well-formed formula that avoids all use of the negation symbol ∼ . (a) Use induction on the length of a wff to prove that if W = W(P1,...,Pn) is...
  14. D

    Faraday's law of mutual induction

    When solving for mutual induction, how do you know what surface to take the flux over?
  15. A

    MHB How Do You Prove a Recursive Sequence Using Strong Induction?

    Hi, I'm currently having trouble with using strong induction to prove a recursive sequence and don't know where to begin, any help would be great thanks. Define a recursive sequence f(0), f(1), f(2),... by f(0) = 0 f(1) = 1 f(n+1) = 3f(n) + 10f(n-1), for all integers n>=1 Prove by strong...
  16. R

    Engineering Characteristics in three-phase induction motor

    Homework Statement I can not understand this question: "Describe the characteristics of a three-phase induction motor?" Does it mean speed,torque,etc? Thanks Homework Equations The Attempt at a Solution
  17. D

    Electromagnetic induction (production of emf)

    Homework Statement the ans is C for this question. i really have no idea , van someone explain please? Homework Equations The Attempt at a Solution
  18. D

    Electromagnetic induction graph of E versus I

    Homework Statement for this question, i only know that E=-L dI/dt... the ans is C , can anyone explain how to get C? Homework Equations The Attempt at a Solution
  19. evinda

    MHB Proving Induction Formula for C(n,k) Using Combinatorial Relations

    Hello! :) Knowing that $\forall k,n \in \mathbb{Z}_{\geq 0} : C(n,0)=1, C(n,k)=0 \text{ for } k>n, C(n,k)=C(n-1,k)+C(n-1,k-1) \text{ for } 1 \leq k \leq n$ show that $\forall 1\leq k \leq n, n \in \mathbb{N}, C(n,k)=\frac{n!}{k!(n-k)!}$ That's what I have done: For $n=1...
  20. O

    Mathematical Induction questions

    Are these satisfactory answers? Question 1: http://i.imgur.com/EElOqwM.jpg Question 2: http://i.imgur.com/SlE2jza.jpg?1 Thank you :)
  21. M

    Question on Electromagnatic Induction

    Homework Statement 9 volt battery 1 Nail Few paper clips Cooper Wire Switch Homework Equations Why my Electromagnetic Induction experiment failed ? It did not create electromagnetic field to attract surrounding paper clips. What are missing here ? How to do successful home-made...
  22. E

    Why do we consider induction a way of charging?

    I get that by getting a charged object near an uncharged one, charges of the opposite sign are induced on the surface, but, since the total charge of the whole object is still zero (because electrons move inside the conductor and can't escape because surroundings insulate it) , the object as a...
  23. K

    Can the Well-Ordering Theorem Prove the Axiom of Induction?

    The well ordering theorem states that every non-empty set has a least element for some ordering (<). This means that if we take the set of natural numbers, by considering the Peano Axioms ONLY, we can find an order (<) (not necessarily the usual one) in which the set of natural numbers N has a...
  24. N

    High resistance or low resistance for induction cooktops?

    Induction heating relies on an ac current to produce a changing magnetic field and hence a conductor will experience a changing magnetic flux. This induces an emf and then a current and thus heat. Heat generated is P=I^2 * R From this, it would appear that a very high resistance would lead...
  25. A

    Is the Sequence an+2=an+1+an Monotonically Increasing?

    Homework Statement Prove that an+2=an+1+an where a1=1 and a2=1 is monotonically increasing. Homework Equations A sequence is monotonically increasing if an+1≥an for all n\inN. The Attempt at a Solution Base cases: a1≤a2 because 1=1. a2≤a3 because 1<2. Am I supposed to prove...
  26. AakashPandita

    Charge produced at end of an insulator rod using electric induction

    When a charged body is brought near one end of a rod, opposite charge is produced at the opposite end of the rod. This is written in my book. Why is charge produced at only the opposite end?
  27. C

    MHB Proof by Induction Help: Solving Recursive & Closed Formulas

    I am having a heck of a time with this proof. I am given both the recursive and closed formula and am supposed to prove that they are equivalent using induction. This is what I am given: bk = 3bk-1+2 for k <2 and b1=2 the closed formula is bn= 3n-1Trying to solve it this is as far as I have...
  28. S

    MHB Proving Induction Principle using Well Ordering Principle of Natural Nos

    In proving the induction principal using the well ordering principal of the Natural Nos the following proof was suggested: Given : 1) Let S be a subset of N (=Natural Nos) 2) $1\in S$ 3)$\forall y[y\in S\Longrightarrow (y+1)\in S$ Then prove : S=N Proof: Let , T= N-S and $T\neq\emptyset$...
  29. MarkFL

    MHB Parrot Guy's question at Yahoo Answers regarding a summation proof by induction

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  30. P

    Induction, proving function divisible by 8

    Homework Statement Prove if n is an odd integer then n^2 - 1 is divisible by 8 The Attempt at a Solution I already set up the whole inductive step and everything but I am trying to prove it works for n+1, here is what I have so far... n=2k+1 ((2k+1)+1)^2 - 1 = 4k^2 + 8k + 3 = 8m Where k...
  31. P

    Induction Motor Torque Decreasing when Load is increased

    Hi. Having measured an induction motor on no load and obtained a torque figure, I have noticed that when load is increased, the torque continues to decrease. How it this possible as I am observing a reduction in rotor speed and increase in rotor currents and current drawn from supply?
  32. P

    Induction Motor Slip decrease with frequency decrease for same load.

    Hi, Does anyone know or could help me understand the reasoning for the amount of slip within an induction motor decreasing with frequency under the same load conditions? I.e. For no external load, the slip is higher at 50Hz than at 20Hz etc? Thanks
  33. P

    Induction Motor Query: Current, Power, Torque & Speed

    Hi, I am currently simulating a 3 phase induction motor system. One of the tests that I have run is to keep the motor parameters (reactance, resistance etc.) constant, and then change the frequency of the supply (50, 40, 30 Hz etc.) before measuring input parameters (current and power drawn...
  34. P

    Electromagnetic Induction Problem

    Homework Statement A 425 mA conventional current is induced in a 48.0cm wire moved through a magnetic field. A 11.5 Ω resistor is connected to the wire as shown in the diagram below. If the magnetic force acting on the wire is 323 mN, then determine the direction and magnitude of the velocity...
  35. M

    Proof by induction - fractions

    Homework Statement I have been working on this proof for a few hours and I can not make it work out. $$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$ i need to get to $$1-\frac{1}{k+2}$$ I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$ then I have tried...
  36. M

    Proof by induction - fractions

    Homework Statement I have been working on this proof for a few hours and I can not make it work out. $$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$ i need to get to $$1-\frac{1}{k+2}$$ I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$ then I have tried...
  37. M

    Proof by Induction Homework: Solve P(n)

    Homework Statement I have two problems that I can't figure out how to turn into summations so I can solve them. 1. Prove by induction (\foralln)P(n), P(n) is 3|(4^{n}-1) I believe the | means divides. 2. Prove by induction (\foralln)P(n), P(n) is n! \leqn^{n}
  38. M

    Proving (n+1) Sum of Squares with Induction

    Homework Statement Prove by induction that the sum of the first n "squares" is n(n+1)(2n+1)/6 In other words prove (\foralln) P(n) where P (n) is \sum^{n}_{i=1}¡^{2}=\frac{n(n+1)(2n+1)}{6} Homework Equations This is just not clicking for me right now. I have no idea if i am just epic...
  39. P

    Induction motor rotor and stator fluxes

    Hi, When a load is applied to the rotor of an induction motor, the rotor speed decreases and as a result, the induced currents in the rotor and the torque produced increases due to the relative speed difference between the rotor and the rotating field within the stator. My question is...
  40. D

    Faraday's law of induction with ferromagnets?

    An electromagnet pulls a ferromagnet from a distance(r), now there is a change in flux. The total magnetic field in this case is increases or decreases? From my understanding the alignment of the domains should increase the magnetic field, but it is not noticeable... Now since there is Δ in...
  41. T

    Proof by Induction involving Binomial Coefficients

    Homework Statement Prove by induction that for any positive integers a, b, and n, (a choose 0)(b choose n) + (a choose 1)(b choose n-1) + ... + (a choose n)(b choose 0) = (a+b choose n) Homework Equations (x choose y) = (x!)/((x-y)!y!) The Attempt at a Solution I am able to do the...
  42. S

    How to Prove a Sequence by Induction

    Define the sequence of integers a1, a2, a3,... as follows: a1 = 3 a2 = 6 an = 5an-1 - 6an-2 + 2 for all n ≥ 3 Prove that an = 1 + 2n-1 + 3n-1 ------------------------------------------------------------------------------------------------ my attempt: base case: n=1 1+ 20 +30 = 1...
  43. L

    How Do You Sum a Series Using Sigma Notation and Identities?

    Homework Statement (5^2) + (11^2) + (17^2) +...+ (18n-1)^2 a)Write the sum in sigma notation b)Using the sigma identities solve the sum (easy to do)Homework Equations ∑i = .5*k*(k+1) etc The Attempt at a Solution The problem I'm having is with the 25 and 121. I thought it was ∑ (6n-1)^2...
  44. B

    Polynomial roots & Mathematical induction

    hi i have this homework question and I am not sure if my thought process is valid. The Question: let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n) now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt: ------------- first off...
  45. C

    Can someone tell me if conduction is more powerful than induction?

    Homework Statement In the lab I am writing about, here were the directions 7. Predictions: Touch a metal sphere to neutralize it. Holding the plastic handle, bring the metal sphere in contact with the charged foam slab. Do not touch the sphere at any time. Take the sphere away from the...
  46. kira506

    Mutual Induction: How is Current Transferred?

    How is the back induced current in the secondary coil transferred to the primary ? I know how mutual induction occurs (and I know that a the bst possible example would be a transformer) but I don't understand how it actually occurs ( I know it,but I still can't get it) I mean I get why when...
  47. Entanglement

    Is back EMF equal to the induced EMF in self-induction?

    In the case of self induction, when the intensity increases gradually an emf is induced which generates a back current " equal in magnitude, opposite in direction " to oppose the increase, so if that is right shouldn't the intensity in the coil Become ZERO till the intensity becomes steady?
  48. MarkFL

    MHB Proof of Product Formula by Induction: Ruslan's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  49. D

    Generalized mathematical induction

    Recall that the fibonacci sequence is defined as { f0=0; f1 = 1 and {fn = f n - 1 + fn -2 for n 2 Prove by generalized mathematical induction that fn = 1/sqrt(5)[ϕn - (-ϕ)-n] where ϕ = [1+sqrt(5)]/2 is the golden ratio.. (This is known as de Moivre's formula.) So...
  50. T

    Induction Motor Speed: Electrical vs Mechanical

    Hello Everybody, Can anyone explain, what is thee difference between 'electrical spped' and 'mechanical speed' of rotor in an induction motor? what do they signify? Many literatures say that electrical speed is no of pole pairs times mechanical speed, if somebody can explain... Thanks
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