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.
Hi physics forum,
I have no idea where to start with this:
As far as I know the general pattern for this sort of proof is,
1) All atomic well-formed formulas (wffs) have some property P
2) From the assumption that immediate predecessors of any non-atomic wff A have P, so too does A...
Hi physics forum,
I have no idea where to start with this:
As far as I know the general pattern for this sort of proof is,
1) All atomic well-formed formulas (wffs) have some property P
2) From the assumption that immediate predecessors of any non-atomic wff A have P, so too does A...
Hello!
I wanted to know if it is possible to calculate the rotor slip if I know the output torque and rpm shaft. I also know the equivalent cicuit of the motor: Rs, Rr, Ls, Lm and Lr.
Thank you!
I was wondering whether an induction hob cooker would work in reverse if a hot pan is placed on the coil - so would this generate a current? I would assume not as the heat induction pan wouldn't be able to create a magnetic field in the coil?
Homework Statement
Prove by induction that no matter how one chooses a set of n+1 positive integers from the first 2n positive integers, one integer in the set divides another integer in the set.
2. The attempt at a solution
Tried direct induction. Base case easy to prove. P(n+1) is with n+2...
Homework Statement
Prove that P(n,m) m+n = n+m for all m,n in natural numbers.
Homework Equations
The Attempt at a Solution
I prove by induction.
Base case: P(0,0) = 0+0 = 0+0.
Inductive step: Let n be an arbitrary natural number. Suppose m+n =n+m. Adding 2 to both sides of...
Let the number x_{n} be defined as follows x_{1} = 1 and x_{2} =2 and x_{n+2}=\frac{1}{2}(x_{n+1} +_{n}) for all n \epsilon N. Use the principle of strong induction to show that 1\leqx_{n} \leq2 for all n \epsilon N
I have never done strong induction before so is this right?
Pf/
so P(1) is...
Homework Statement
I need to prove that for any real number r, if 0 < r < 1, then for all positive integers n and m, if n < m, then r^n > r^m.
Homework Equations
No calculus techniques are permitted, only mathematical induction.
The Attempt at a Solution
I know that any...
Homework Statement
Q. Prove by induction that... (please see attachment).
Homework Equations
The Attempt at a Solution
The end result should be divisible by 6, but hasn't worked out for me. Can someone help me spot where I've gone wrong? Thank you.
Hi,
I've been thinking about the shape of the voltage waveform induced by a magnet falling through a coil. I know (both intuitively and from empirical experience) that the voltage should become increasingly positive as the magnet approaches the coil, then it should decrease rapidly...
Homework Statement
Prove by finite list induction
∀xs:: [a]. map (\x -> x) xs = xs
Homework Equations
map f [ ] = [ ]
map f (x : xs) = (f x) : map f xs
The Attempt at a Solution
The list induction principle to prove a finite lists of type a is
[P([ ])^
∀x :: a. ∀xs ...
Homework Statement
I like this thing call induction
Prove that for every integer n\geq 0 that
5| (9^n - 4^n)
The Attempt at a Solution
1) Base Case n = 0
9^0 - 4^0 = 0
Clearly it is true
2) Take n = k + 1
9^{k + 1} - 4^{k + 1} = 9^k 9 - 4^k 4 = 9^k (4 + 5) - 4^k 4...
Hello,
I have a question about inductive reasoning...
Earlier this week my intro proofs class went over the logical structure of induction, and an example.
The example was a proof of \Sigmai = n*(n + 1)/2
My main issue is the assumption that "p(k)" is true. What if it's not? I asked this in...
Homework Statement
Prove that weak induction is equivalent to strong induction.
Homework Equations
To prove this...we assume weak induction. That is
(i) 1 \in S
(ii) n\in S \Rightarrow n+1 \in S for all n in N
The Attempt at a Solution
So to prove strong induction we...
Homework Statement
Prove that for any positive h and any integer n\geq0, (1+h)n\geq1+nh+\frac{n(n+1)}{2}h2.
Homework Equations
None.
The Attempt at a Solution
I proved that P(0) is true (1\geq1). The rest of the proof goes as follows:
Assume K\inZ (the set of integers) and P(K)...
Homework Statement
Prove that 1/(1-x) = 1 + x + x2 + x3 + ... + xn/(1-x) for n>=2
Homework Equations
The Attempt at a Solution
I'm not really all that sure how to begin. The base case would be 1/(1-x) = x2/(1-x) and the induction hypothesis would be 1/(1-x) = 1 + x + x2 + x3 +...
What will happen if an ac inductions motor is running, an suddenly an error occurs in the voltage pr. hertz motor control so the stator frequency goes down below the rotor frequency for 1-2 sec?
Hi,
So, I was assigned a problem in my Intro Analysis course that involves proving, by induction, that the set A minus some arbitrary number of intersections of the sets B_{j} is equal to some arbitrary number of unions of A minus the sets B_{j}.
I've written out a proof, but I'm not too...
consider first , an induction motor... a rotating magnetic field is applied to an electrically conductive rotor, and the eddy currents induced will oppose the change in magnetic field experienced by the rotor, hence the rotor gets torque
in an induction motor the rotating magnetic field is...
Homework Statement
1b) Prove by induction: 1^{3}+...+n^{3}=(1+...+n)^{2}
2a) Find a formula for: \sum^{n}_{i=1}(2i-1)
Homework Equations
There's a Hint for 2a): 'What to this expression have to do with 1+2+3+...+2n?'
The Attempt at a Solution
In 2a) I've got near the answer...
I'm having a bit of trouble understanding why the principle of induction is included as one of Peano's axioms. It seems like it should not be independent of the others. Obviously it can be stated as:
If a predicate P is true only of natural numbers, P(0) is true, and also P(n)\rightarrow...
Hi
I'm trying to preform a simple model of a ferromagnetic bar which has a coil wrapped around it. The bar has a 4×4 cm base and is 15 cm tall, the coil has an inside radius of 29 mm and outside radius 30 mm and a height of 10 cm. The goal is to find the eddy currents in the bar, generated...
If I wanted to know how to find the magnetic flux through a loop, I'd need to know the current flowing near it. If that current is in the form of a spark, how do I represent that?
The spark is nearly instantaneous, and I've got no idea how to determine the number of charges which flowed...
Homework Statement
suppose we have a coil sitting on a desk (with radius =1), and then a spark passes by the side of it, in such a way that the magnetic flux through the coil is maximized. The coil is attached to a full wave rectifier which is connected to a very large capacitor.
There is an...
Hi. I am learning mathematical induction on my own and I came across this problem:
Homework Statement
Prove:
1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)²2. The attempt at a solution
Quick test for n=1:
(3 -2)(3 + 1) = 1(1 + 1)²
4 = 4
Alright, so I rewrite this with, on the left...
Homework Statement
If i want to use induction to prove the Fibonacci sequence I first check that 0 satisfies both sides of the equation. then i assume its true for n=k then show that it for works for n=k+1
The Attempt at a Solution
But I am a little confused if i should add another...
new guy here with a simple magnetic theory question.
if i were to coil 10 turns of insulated wire around an 1/8" round x 1" long neodymium magnet, then wire it in parallel to a second magnet with 20 turns of the same gauge wire, what kind of change could i exspect in the second magnets field...
Homework Statement
Use the Principle of Mathematical Induction and the Product Rule to prove the Power Rule when n is a positive integer.
Homework Equations
Dxxn = nxn-1
Dx(fg) = fDxg + Dxfg
The Attempt at a Solution
In summary,
Dxxn = nxn-1
Dxxk = kxk-1
Dxxk+1 = (k+1)x(k+1)-1
Dx(xkx) =...
The resistance per unit length of a conducting wire is proportional to the square root of the ratio of permeability to conductivity.
The power generated as heat may be expressed as I-squared x R: also as V-squared / R.
In induction, the EMF induced is determined by the rate of...
I'm trying to model a vertical axis wind turbine with the double-multiple streamtube model (with Matlab), where wind speed is dampened by an induction factor so that the output is different for upwind and downwind parts of rotation.
The trouble is that for the downwind part, the induction...
Homework Statement
Let n be a natural number.
Prove that 0!+1!+2!+...n! < (n+1)!
and my < sign should be less than or equal to.
The Attempt at a Solution
The proof is by induction
it works for 0 because 0! is less than or equal to 1!
now we assume it works for n=k by the induction...
Homework Statement
Here are a bunch of questions that I've attempted. I feel fairly confident with my answers, but I just wanted to make sure :) Thanks!
a) calculate the synchronous speed, amount of slip and the percentage slip in a typical aircraft a.c., four pole, 400hz induction motor...
Fig.35 shows a light and flexible conducting loop X freely hung on a smooth horizontal rail. A
bar magnet PQ approaches the loop from the right. Which one of the following descriptions
about this process is correct?
A If P is a N-pole, the loop will be repelled to the left and its area...
Homework Statement
Check out the attached image.
The electromagnetic field B is constant.
The rod is moving through the rail and an inductive emf V is produced on the closed loop.
Due to that, the condenser C has a voltage difference Vc.
Question:
Express the relation between the...
Homework Statement
Prove and show that 2n ≤ 2^n holds for all positive integers n.
Homework Equations
n = 1
n = k
n = k + 1
The Attempt at a Solution
First the basis step (n = 1):
2 (1) ≤ 2^(1) => 2 = 2.
Ergo, 1 ϵ S.
Now to see if k ϵ S:
2 (k) ≤ 2^k
But, k ϵ S implies k...
Homework Statement
n2<=2n
n is a natural number
For what values of n is the statement true and prove by induction.
Homework Equations
The Attempt at a Solution
I tried 1 and it worked, I tried 2 and it worked, just for fun I tried 3 and it didn't work, so I assumed the...
Homework Statement
Prove:
1^3 + 2^3 +...+n^3=(1+2+...+n)^2
n=Natural number
Homework Equations
The Attempt at a Solution
Using induction -
n=1 obvious
Assume for n=k equation is true.
Show for k+1.
I have that the right side prior to k+1 is (k^2(k+1)^2)/4
After k+1 I...
Hello, I have come to this forum as I don't believe what i am being taught to be entirley accurate and need some clarity on a question (not an exact answer, this is not homework)
To begin, I believe the person 'teaching' my subject is not up to standard as when our final topic of three...
NOTE: This is not a homework problem.
This semester we dealt with AC motors both three phase and single phase.
Now I know that 3 phase system is associated with constant power and therefore a constant power is being fed into that motor (and hence converted into constant mechanical power)...
I was doing this for practice and came across this problem-- I have no clue how to prove it.
By Induction, prove that 3n-1 is divisible by 2.
1) 3-1=2, divisible by two; good so far
Now I have no clue how to approach this.:bugeye:
Please help. Thanks in advance:smile:
Can anyone explain why cranes in construction sometimes experience induced currents from antennas radiating from distances of up to several Kilometers.
I have looked into the phenomenon several times, and have not been able to find a study or baselines to follow. I have a basic understanding...
Can anyone explain why cranes in construction sometimes experience induced currents from antennas radiating from distances of up to several Kilometers.
I have looked into the phenomenon several times, and have not been able to find a study or baselines to follow. I have a basic understanding...
Homework Statement
Suppose an/n+1 +...+a0/1=0.
Prove f(x) =anxn +...+a0 has a root between zero and one.
Homework Equations
I'm pretty sure this is induction, but I'm not completely sure.
Mean Value Theorem probably
The Attempt at a Solution
Well f(0)=a0 and f(1)=an + ... +...
Homework Statement
Prove that
11^n - 4^n
is a multiple of 7
Homework Equations
N/AThe Attempt at a Solution
I substituted k+1 in for n and simplified to get
11(11^k)-4(4^k)
but after this point I get stuck. Any help would be appreciated.
Homework Statement
Summation of i(i + 1) (with i going from i = 2 to i = n-1) = n(n-1)(n=1) / 3
a. Write P(2). Is P(2) true?
b. Write P(k)
c. Write P(k+1)
d. Prove by mathematical induction that the formula holds true for all integers
n \geq 2
Homework Equations...