What is Induction: Definition and 999 Discussions

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

    Proof by induction for rational function

    For this problem, My solution is ##P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0## where n is a member of the natural numbers Base case (n = 1): ##P(x) = a_0x^0 = a_0## Thus ##\lim_{x \to \infty} \frac{P(x)}{e^x} = \lim_{x \to \infty} \frac{a_0}{e^x} = a_0 \lim_{x \to \infty}...
  2. AntonAv

    Induction heat flux density induced by wire in a slab

    TL;DR Summary: How to calculate induction heat flux (density) induced by an infinitely long wire in an infinitely large slab. Hi, I have to simulate induction heating caused by a straight long wire in a thick slab of material (no strict limitations). To make it, in the best case I should...
  3. I

    Calculate current through tank circuit on induction heater

    I want to measure tank circuit currents using *different* methods. Below is the tank circuit. The coupling transformer has 20 turns and has 10A @ 80khz going through it. This means there should be a maximum of 200A in the tank (20:1 turns ratio). I measure the voltage across the 2.6uF capacitor...
  4. P

    I Help with basic transfinite induction proof

    Some definitions: The following statement has been left as an exercise in transfinite induction in a handout. I'm looking at Wikipedia and am trying to follow their outline: 1. Show it for the base case, i.e. that ##\mathcal{F}_{0}\subset\mathcal{G}##. This is, however, trivial, since we...
  5. A

    Number of Multiplications in the FFT Algorithm

    Hello everyone, maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this? I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
  6. MatinSAR

    Ways of transferring charge between objects (conduction and induction)

    Two ideas: 1) Negative charged object get close to A then we close K1 key. So -10SI charge from sphere A moves to sphere B and 30SI charge moves from sphere B to sphere A. At this moment sphere A has 30SI charge and sphere B has -10SI charge. 2) We close K1 key so system of 2 spheres should have...
  7. L

    Induction with binomial coefficient

    Hi, I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}## I proceeded as follows: $$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
  8. M

    A Question About Induction: Faraday's Law versus Motional EMF

    Visualize a rectangular bare wire circuit with one side loose so that it can slide along the adjacent sides and thereby change the size of the circuit. There is a static magnetic field orthogonal to the plane of the circuit and linking the circuit. There will be a motional emf in the moving...
  9. E

    I Is total flux linkage λ=dΨ*Ienclosed/I or λ=N*Ψ?

    In Sadiku, he used the formula dλ=dΨ * Ienclosed/I to determine the total flux linkage for coaxial cable for ρ<a and for a<ρ<b, but I applied this formula for the solenoid and it didn't work, the way that works for the solenoid is by using λ=N*Ψ. So why we multiply by Ienclosed/I in the coaxial...
  10. L

    Does a Bent Wire Affect Induction Calculations?

    Hi, unfortunately, I am not sure if I have calculated the task correctly here Task a I have now proceeded in such a way that I thought that the magnetic field only flows through the area drawn in red. Which ##\frac{1}{4}## corresponds to the area of a circle. By the fact that the magnetic...
  11. E

    Inducing Voltage in a Transformer: How Does It Work?

    A changing current in a transformer primary produces a changing magnetic field, which induces a voltage in the secondary (correct?), but if no circuits are closed on the secondary, there's no current in the secondary (and therefore primary as well). So how is this voltage induced?
  12. Grizzly_1

    B Confusing diagram of a rotating coil in a magnetic field

    Hello all, I am currently studying for a physics a-level qualification in the UK, I use the AQA specification and I am having trouble understanding this image representing a scenario I found in my textbook. The first image in the three part diagram shows this rotating coil and to me, it makes...
  13. nomadreid

    I Prove Even # Transpositions in Identity Permutation w/ Induction & Contradiction

    There is a proof that shows by induction (and by contradiction) that the identity permutation decomposes into an even number of transpositions. The proof is presented in the first comment here...
  14. J

    Best way to measure magnetic waves for electromagnetic induction

    Hi all, Looking to measure some magnetic waves being generated at an electric coil. Freq is between 0-20kHz and magnitude is pretty small <1T. Any have suggestions for the best tool to measure and log data of this magnetic waveform? Googling around, I found meters like this: [Possible spam...
  15. K

    Induction heater using Bicycle Dynamo Generator @ 24 V 60HZ Supply

    I was wondering about the possibility of Induction heater powered by Bicycle Dynamo (alternator/generator) without using any other circuit element. Which means connecting 24 V 50 to 60 HZ AC to Coil with turns around 100 in pancake shape (10 inch diameter ) and Stainless steel plate of 3 mm...
  16. G

    Building a Safe and Efficient Induction Heater Circuit: Tips and Tricks

    If you have not built an induction heater yet it is a fun project. Build the small circuit with 6 yellow capacitors first. Induction heater is a fun project. I built a small induction heater then I wanted a more powerful one. A small unit is so simple you can build it in about 2 hours work...
  17. K

    Induction Heater at 50Hz or 60Hz from Kitchen Wall Outlet.

    I'd like to make an Induction Heater design for Cooking purpose, where heat can reach up to 400-500F or so. I read in some other forums, that it's possible. So I'm trying to get more ideas and possibly learn how this can be done safely (in case I make it / if it's really safe). The Idea is use...
  18. abdulbadii

    Transformer frequency contradiction on induction heating

    As a transformer freq bet higher, inside induction get more efficient i.e. less loss: 1. Hysteresis loss = η * Bmax^n * f * V. 2. Eddy current loss( proportional to B2mf2Bm2f2 )Now it seems that losses increases with increase in efficiency... But the above equations are valid when max flux...
  19. A

    Induction motor rotor design features

    I had to repair (change bearings) on a single phase induction motor recently and I recalled an age old question I've had. What are the rotor end ring blades made for ? The only reasonable answer I can come up with is - they are for cooling of the stator winding ends that extend out of the...
  20. chwala

    Show the proof by induction in the given problem

    My interest is solely on the highlighted part in red...hmmmmmmm :cool: taken a bit of my time to figure that out...but i got it. Looking for any other way of looking at it; I just realised that the next term would be given by; ##\dfrac{1}{4}(k+1)^2(k+2)^2-\dfrac{1}{4}k^2(k+1)^2##...
  21. chwala

    Prove by induction or otherwise, that ##\sum_{r=1}^n r(3r-1)=n^3+n^2##

    My take; ##\sum_{r=1}^n r(3r-1)=n^3+n^2## Let ##S_n=\sum_{r=1}^n r(3r-1)## and ##f(n)=n^3+n^2## then, ##S_1 = 2## and ##f(1)=2## hence the result is true when ##n=1## Assume that ## S_m = f(m)## for some integer ##m## then, ## S_{m+1}= (m+1)(3(m+1)-1) +S_m ## ## S_{m+1}=...
  22. Argonaut

    [L'Hospital's Rule] Can I Use Mathematical Induction to Prove This?

    The Student's Manual simply applies l'Hospital's Rule n-times to arrive at ##\frac{e^x}{n!}\to\infty## as ##x\to\infty##. However, I'm wondering if I could use Mathematical Induction to prove this. Is the following correct and sufficiently rigorous (at least for an undergraduate-level Calculus...
  23. Pipsqueakalchemist

    DC motors and Induction motor: What does "no load conditions" mean?

    TL;DR Summary: So I'm just confused when the question asks me to solve for the no load speed of DC motors and induction motors. Does no load condition mean that the output torque (Tout) is zero? This is what I was assuming so far for both DC and induction motor. Is no load condition the same...
  24. Lotto

    How to calculate magnetic induction just beyond the end of the magnet?

    I know I should use a limit $$B=\lim_{r\to l}{\frac{\mu}{4\pi}\cdot \frac{4rml}{{\left({r}^2-l^2\right)}^2}},$$,but in Wolfram I get a weird solution. https://www.wolframalpha.com/input?i2d=true&i=Limit[Divide[4rl,Power[\(40)Power[r,2]-Power[l,2]\(41),2]],r->l] What is the solution? It...
  25. murshid_islam

    I Is there any way to simplify my proof by induction?

    I'm trying to prove the statement ##n^2 + 1 < n!## for ##n \geq 4##. My proof by induction looks way too contrived. Is there a way to simplify it? Here's what I got. For n = 4, ##n^2 + 1 = 17 < 4!##. So, the statement is true for n = 4. Now let's assume it's true for n = k, that is, ##k^2 + 1 <...
  26. A

    Prove by the principle of induction

    (expression given to be proven) check for p(1)... 2=2 substitute (n+n) to And here is the problem, I just can't find a way to continue solving this problem
  27. C

    Prove by induction the sum of complex numbers is complex number

    See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
  28. A

    Series inequality induction proof

    My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}## then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}## We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for...
  29. M

    I Maxwell-Faraday Law & Stealth Magnets: Exploring EMF Induction

    Consider a closed path consisting of a loop of wire with a nonconducting gap that completes the closed path. The wire is threaded through a toroidal permanent magnet, magnetized around the toroid (what I call a stealth magnet). The magnetic flux is considered to be confined to the magnet. The...
  30. FFXT

    I Faraday induction in constant B field, with non-conduction wires

    A standard textbook problem features a constant B field and a conducting loop that increases in area at constant rate. It is easy to work out the induced EMF and the associated electric field magnitude and direction (CW or CCW). The magnitude of the E field is E = B v where v is a velocity...
  31. J

    Induction Proof, Apostol Calc Vol I, I.4.4.7

    For calculating ##n_1## I had no problems as ##n_1 = 3##. PF: Show ##(1+x)^n > 1 + nx + nx^2 ## is true for all ##x \ge 3##. Let ##n = 3, (1+x)^3 > 3x^2 + 3x + 1 ## ##x^3 +3x^2 +3x + 3 > 3x^2 + 3x + 1 ## is true. Assume ## n = k ## such that ##(1+x)^k > 1+kx + kx^2, k \ge 3## is true. We...
  32. benorin

    This is for an Insights article: Bivariate induction proof using Calc3

    Link to my insight Article it's right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of...
  33. chwala

    Prove this problem that involves Mathematical induction

    Ok, would i be correct to approach it this way, Let ##n=1##. If ##n=1##, then ##5^1+3## is divisible by ##4##, the statement is true for ##n=1##. Assume its true for ##n=k## ∀ ##kε\mathbb{z}^{+}.## Then ##5^k+3## is divisible by ##4.## i.e ##5^k+3=4m## ∀ ##m ε\mathbb{z}^{+}## Let ##n=k+1.##...
  34. H

    I Proof of Induction Principle starting from N, not from 1

    In this video lecture (though I have linked the video at "current time", in case it doesn't; work please see the video at 19:16), the lecturer just works out (he is not explaining anything) the proof of Induction Principle starting from ##N##. Let me give out here what he did: Statement: Let N...
  35. M

    Use induction and the Bertrand conjecture to show that....

    Proof: The proof is by induction. (1) When ## n=4 ##, the statement is ## p_{4}<p_{1}+p_{2}+\dotsb +p_{3} ##, which is true, because ## 7<10 ##. (2) Now assume ## n=k+1 ##. Then ## p_{k+1}<p_{1}+p_{2}+\dotsb +p_{k+1-1}\implies p_{k+1}<p_{1}+p_{2}+\dotsb +p_{k} ##. Thus ##...
  36. billyt_

    Experimental investigation of Faraday's Law of Induction

    Looking at how the induced EMF is proportional to the rate of change of magnetic flux, intuitively it seems that if I increase the velocity of the magnet through the solenoid, i.e. drop it from a higher height, the EMF should increase as well. However, I am unsure if this is true and can't seem...
  37. docnet

    Prove by induction the inequality

    The statements holds for the case ##n=1## \begin{align} 1+x \leq & 1+x+\frac{1}{2}\cdot 0\cdot x^2\\ =&1+x \end{align} Assume the statement holds true for ##n=k## $$(1+x)^k\geq 1+kx+\frac{k(k-1)}{2} x^2$$ Then, for ##n=k+1##, we have the following \begin{align} (1+x)^k\cdot (1+x)\geq&...
  38. Narayanan KR

    Interesting Links Between Faraday's EM Induction and EPR

    Imagine a magnet moving up and down so that its flux 'B' cuts the copper rod to produce an alternating emf, suppose if the movement is fast enough such that its frequency equals to the electron spin resonance frequency given by F = B x 2.8 Mhz per gauss, neglecting skin effect, more copper...
  39. C

    Induction problem: Tk+1 = Tk + Tk-1 + Tk-2

    I have come to the conclusion that the sequence is 1 1 1 3 5 9... trying to do induktion and I get Tp+1=2^p+1, which means that 2^p +2p-1 + 2p-2 <=2^p+1 and this is not really solvable in any satisfactory way. What am I doing wrong?
  40. M

    Engineering Kirchoff's Law Problem about this Induction Generator circuit

    Hi, Please find attached below the Induction Generator circuit. I have the following parameters: Parameter Value Rs = Rr (Resistance on stator and rotor) 0.2 Ω Ls = Lr (inductance on stator and rotor) 5 mH = 0.005 H p (number of pairs of poles) 2 f (frequency) 60 Hz Vsl (stator line...
  41. M

    Engineering Induction motor calculation problems

    Given the following input parameters: Parameter Value Rs (Resistance through stator) 1.4 Ω Rr (Resistance through stator) 0.7 Ω Ls (stator inductance) = Lr (rotor inductance) 0.002 H xs = xr = 2*π*f*Ls 0.6283i Ω Lm(magnetic inductance) 0.01 H xm = 2*π*f*Lm 3.1415iΩ f (frequency)...
  42. Mayhem

    I How do I use induction more rigorously when making Taylor expansions?

    When I do Taylor expansions, I take the first 3 or 4 derivatives of a function and try to induce a pattern, and then evaluate it at some value a (often 0) to find the coefficients in the polynomial expansion. This is how my textbook does it, and how several other online sources do it as well...
  43. M

    Engineering Calculating Power Losses in an AC Induction Motor

    Given the AC Induction (asynchronous motor) in 3 phases: Parameter Value Pn (Nominal Power) = Pmechanical (output power at the shaft) 5 kW = 5000W uls (Voltage through the stator line) 220 V fstator (stator frequency) 50 Hz p (Number of pole pairs) 2 LFe (Iron loss) = Lmechanical...
  44. P

    Using Monopods for city travel utilizing linear induction motors

    This is another open ended question, exploring a space of design concepts, in similar spirit to this. I want to explore monopods with regard to travel in densely populated cities(even possibly intercity travel). The main idea is to use small personalized pods to travel in tubes(or tracks). The...
  45. A

    I Induction heating vs Current heaing

    I've been thinking and can't come up with a satisfying answer. Would there be a difference in the amount of heat generated in a given time in a metal rod if it was heated by a) induction b) current passing through the same length of rod Given that the current is the same amperage and...
  46. C

    Simple Induction Induction proof of Polynomial Division Theorem

    Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...
  47. R

    Multiplication -- prove this one by induction

    Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n The base case, n=0 gives 0 x m++=(0 x m) +0 gives 0=0 Now assume n x (m++) = (n x m) +n For n++ we get n++(m++)=((n++)m) + n++ from this point I am stuck, how can I prove both sides are the same?
  48. Flash27

    I Building a Linear Induction Motor | Tips & Tricks

    I am currently trying to create a linear induction motor for fun and am having some trouble getting it to start oscillating or move at all. I am using this video as a reference... I am using 3D printed PLA as the structure for the copper to wind around, 26 GA Craftware USA copper wire, 5/8"...