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Topic: Mathematical induction

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Proof of mathematical induction

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	NationMaster - Encyclopedia: Mathematical induction (Site not responding. Last check: 2007-11-05)
		Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers.
		Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics.
		Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence.
	www.nationmaster.com /encyclopedia/Mathematical-induction (731 words)

	Mathematical Induction, What is Mathematical Induction
		The use of mathematical induction in demonstrations was, in the past, something of a mystery.
		Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite.
		The principle of mathematical induction might be stated popularly in some such form as "what can be inferred from next to next can be inferred from first to last." This is true when the number of intermediate steps between first and last is finite, not otherwise.
	www.math10.com /en/maths-history/mathematical-induction.html (2721 words)

	Three forms of mathematical induction
		The basis for induction is trivial; the substantial part of the proof goes from case n to case n + 1.
		The basis for induction is vacuously true; the step that goes from case n to case n + 1 is trivial if n ≥ 2 and impossible if n = 1; the substantial part of the proof is the case n = 2.
		The induction step shows that if P(k) is true for all k < n then P(n) is true; no basis for induction is needed because the first, or basic, case is a vacuously true special case of what is proved in the induction step.
	www.ebroadcast.com.au /lookup/encyclopedia/th/Three_forms_of_mathematical_induction.html (173 words)

	Encyclopedia topic: Mathematical induction (Site not responding. Last check: 2007-11-05)
		Mathematical induction is a method of mathematical proof (Proof of a mathematical theorem) typically used to establish that a given statement is true of all natural number (The number 1 and any other number obtained by adding 1 to it repeatedly) s, or otherwise is true of all members of an infinite sequence.
		Note that this form of mathematical induction is actually a special case of the previous form because if the statement that we intend to prove is P(n) then proving it with these two rules is equivalent with proving P(n + b) for all natural numbers n with the first two steps.
		The principle of mathematical induction is usually stated as an axiom ((logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident) of natural numbers, see Peano axioms (additional info and facts about Peano axioms).
	www.absoluteastronomy.com /encyclopedia/m/ma/mathematical_induction.htm (1077 words)

	Mathematical induction - Wikipedia, the free encyclopedia
		Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence.
		This form of mathematical induction is actually a special case of the previous form because if the statement that we intend to prove is P(n) then proving it with these two rules is equivalent with proving P(n + b) for all natural numbers n with the first two steps.
		The principle of mathematical induction is usually stated as an axiom of natural numbers, see Peano axioms.
	en.wikipedia.org /wiki/Mathematical_induction (1098 words)

	tScholars.com \| Mathematical induction (Site not responding. Last check: 2007-11-05)
		Mathematical induction is used to prove that every statement in an infinite sequence of statements is true.
		Using mathematical induction (implicitly) with the inductive hypothesis being that the statement is false for all natural numbers less than or equal to m, we can conclude that the statement cannot be true for any natural number n.
		The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms.
	www.tscholars.com /encyclopedia/Mathematical_induction (2988 words)

	Mathematical proof - Wikipedia, the free encyclopedia
		In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true.
		The distinction has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term).
		The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
	en.wikipedia.org /wiki/Mathematical_proof (584 words)

	Mathematical Induction
		I found that what I wrote about geometric series provides a natural lead-in to mathematical induction, since all the proofs presented, other than the standard one, use mathematical induction, with the formula for each value of n depending on the formula for the previous value of n.
		To appreciate the power of mathematical induction, we will see in the next section how to create a proof for the geometric series formula in a somewhat mechanical manner.
		Actually, it is not unusual in mathematics to come up with a result by intuition or by making assumptions that have not yet been proven to be correct.
	www.geocities.com /Athens/Delphi/5136/Induction.html (1211 words)

	Mathematical Induction
		A type of proof that deserves special attention is mathematical induction.
		This principle of strong mathematical induction is used for example to prove the result that every positive integer is a prime number, a power of a prime or a product of primes.
		As with the principle of mathematical induction, there is a variation on this to cover those statements that are true for all integers beyond a certain value
	www.math.csusb.edu /notes/proofs/pfnot/node10.html (291 words)

	Making Mathematics: Mathematics Tools: Mathematical Induction
		Mathematical induction is a common method for proving theorems about the positive integers, or just about any situation where one case depends on previous cases.
		Mathematical induction and its variations are useful in proving identities that are true for any value of integer, but they do not help you see how someone figured out the identity at first place.
		More information and problems on Mathematical Induction can be found at http://www.math.csusb.edu/notes/proofs/pfnot/node10.html and http://www.cut-the-knot.com/induction.html and in the articles "Teaching Mathematical Induction: An Alternative Approach" and "When Memory Fails" in the September 2001 issue of Mathematics Teacher.
	www2.edc.org /makingmath/mathtools/induction/induction.asp (967 words)

	Mathematical induction - Glasgledius (Site not responding. Last check: 2007-11-05)
		Mathematical induction, or proof by induction, is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.
		It is possible to base a proof of mathematical induction on other mathematical principles.
		Related methods An induction variant is used in computer science to prove that expressions which can be evaluated are equivalent, and this is known as structural induction.
	www.glasglow.com /E2/ma/Mathematical_induction.html (1159 words)

	PlanetMath: an example of mathematical induction
		Below is a step-by-step demonstration of how mathematical induction (the Principle of Finite Induction) works.
		All these steps are essential in any proof by (mathematical) induction.
		This is version 4 of an example of mathematical induction, born on 2006-02-06, modified 2006-06-10.
	www.planetmath.org /encyclopedia/AnExampleOfMathematicalInduction.html (223 words)

	Discrete Algebra - Mathematical Induction
		The principle of mathematical induction is very helpful in proving many statements about positive integers.
		Solution: Apply the two steps of mathematical induction, but prove the equality true for n = 0, since the statement is to be shown valid for all whole numbers (including zero).
		Solution: While mathematical induction can still be applied to prove the above statement, a few changes must be made to the solution process.
	library.thinkquest.org /10030/11matind.htm (762 words)

	Peter Suber, "Mathematical Induction"
		It is like induction in that it generalizes to a whole class from a smaller sample.
		Notice that the induction step is to prove a conditional statement, of which the induction hypothesis is the antecedent.
		Perhaps it does not go without saying that if we are to use mathematical induction to prove that some theorem applies to "all possible cases", then those cases must somehow be enumerable and tightly linked to successive integers.
	www.earlham.edu /~peters/courses/logsys/math-ind.htm (1191 words)

	Mathematical Induction
		Mathematical Induction is way of formalizing this kind of proof so that you don't have to say "and so on" or "we keep on going this way" or some such statement.
		The idea is to show that the result is true for n=1 and then show how once you've shown it to be true for some integer, you can see that it must be true for the next one as well.
		Prove by induction that the sum 1 + 3 + 5 + 7 +...
	bigcheese.math.sc.edu /~sumner/numbertheory/induction/Induction.html (1878 words)

	Mathematical induction - Topics in precalculus
		THE NATURAL NUMBERS are the counting numbers: 1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.
		We have now fulfilled both conditions of the principle of mathematical induction.
		According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove.
	www.themathpage.com /aPreCalc/mathematical-induction.htm (1038 words)

	Math Forum Discussions - Re: Mathematical Induction
		"whence" in mathematical prose (was, Re: Mathematical Induction)
		Re: "whence" in mathematical prose (was, Re: Mathematical Induction)
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?forumID=13&threadID=57013&messageID=221871 (198 words)

	Mathematical Induction
		Mathematical Induction (MI) is an extremely important tool in Mathematics.
		Statements proven by math induction all depend on an integer, say, n.
		Math induction is just a shortcut that collapses an infinite number of such steps into the two above.
	www.cut-the-knot.org /induction.shtml (373 words)

	Logic: Longer Chains of Reason, mathematical induction
		Induction here consists of extracting conclusions from chains of rules and patterns, one after another, perhaps without stopping or end.
		This story leads to the notion called mathematical induction, a method of reason or logic used in mathematics after arithmetic to get conclusions (or climb ladders).
		The principle of mathematical induction stated below describes the above ladder idea in the algebraic shorthand notation favored in mathematics.
	whyslopes.com /etc/ThreeSkillsForAlgebra/ch04.html (809 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: 2007-11-05)
		Another generalization, called complete induction (or strong induction or course of values induction), says that in the second step we may assume not only that the statement holds for n = m but also that it is true for n less than or equal to m.
		Complete induction is most useful when several instances of the inductive hypothesis are required for each inductive step.
		Strictly speaking, it is not necessary in transfinite induction to prove the basis, because it is a vacuous special case of the proposition that if P is true of all n
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Mathematical_induction (2752 words)

	Mathematical Induction Tutorial
		Mathematical induction is a powerful, yet straight-forward method of proving statements whose "domain" is a subset of the set of integers.
		Proof by induction involves three main steps: proving the base of induction, forming the induction hypothesis, and finally proving that the induction hypothesis holds true for all numbers in the domain.
		In the induction hypothesis step, we say that since the statement holds true for at least one value, we can assume that it will hold true for some arbitrary, fixed value of n, usually k.
	www.nipissingu.ca /calculus/tutorials/induction.html (637 words)

	NSW HSC ONLINE - Mathematics
		Students, especially those in Year 11, often find the basic idea of mathematical induction a little confusing and are convinced that the teacher has cheated.
		Parallel this mathematical concept with methods used in science, where experiments are done, a hypothesis is formed, and then more work is done to confirm the hypothesis.
		Explain the method of mathematical induction and use it to prove this formula.
	hsc.csu.edu.au /maths/ext1/math_induction/179/mathinduc.htm (668 words)

	mathematical induction from FOLDOC (Site not responding. Last check: 2007-11-05)
		Roughly, in the basis the theorem is proved to hold for the "ancestor" case, and in the induction step it is proved to hold for all "descendant" cases.
		The most common instance of proof by induction is induction over the natural numbers where we prove that some property holds for n=0 and that if it holds for n, it holds for n+1.
		induction) for an arbitrary case k, or that it holds (in strong mathematical induction) for all cases up to and including k.
	www.swif.uniba.it /lei/foldop/foldoc.cgi?mathematical+induction (280 words)

	Mathematical induction - Definition, explanation
		A somewhat more general form of argument used in mathematical logic and computer science shows that expressions that can be evaluated are equivalent; this is known as structural induction.
		Another generalization, called complete induction, allows that in the second step we assume not only that the statement holds for n = m but also that it is true for n smaller than or equal to m.
		Suber, explaining the difference between inductive inference and mathematical induction (which is a species of deductive inference).
	www.calsky.com /lexikon/en/txt/m/ma/mathematical_induction.php (1154 words)

	Mathematical Induction (Site not responding. Last check: 2007-11-05)
		Mathematical induction: The two-phase process of proving a theorem involving a natural number.
		Mathematical induction is actually a "deductive complement" to the induction process that presumably yielded the theorem in the first place.
		Conclude that the theorem is true for all n by mathematical induction.
	jwilson.coe.uga.edu /emat4500/Induction/MathematicalInduction.html (450 words)

	CS312 Mathematical Induction (Site not responding. Last check: 2007-11-05)
		In all inductive proofs, it is good practice to state the induction hypothesis explicitly and indicate explicitly where you use it in your proof that P holds for n.
		In a proof by induction, we first show that the statement is true for the "smallest" values of our inductively defined set, which for the natural numbers is just 0.
		Induction is often used to show that a recursive procedure computes the correct answer.
	www.cs.cornell.edu /courses/cs312/2001sp/handouts/induction.html (1496 words)

	Mathematical induction (Site not responding. Last check: 2007-11-05)
		Mathematical induction is a proof method, used to prove properties of objects that exhibit certain regularities.
		Mathematical induction is applicable to objects that have some regularities and can be ordered.
		Then, by the principle of mathematical induction it will follow that the property is true for all objects.
	www.simpson.edu /~sinapova/cmsc180a/L13-Induction.htm (1079 words)

	Mathematical induction (Site not responding. Last check: 2007-11-05)
		Asomewhat more general form of argument used in mathematicallogic and computer science shows that expressions that can beevaluated are equivalent; this is known as structuralinduction.
		Note that thisform of mathematical induction is actually a special case of the previous form because if the statement that we intend to proveis P(n) then proving it with these two rules is equivalent with proving P(n + b) forall natural numbers n with the first two steps.
		The principle of mathematical induction is usually stated as an axiom of naturalnumbers, see Peano axioms.
	www.therfcc.org /mathematical-induction-33158.html (735 words)

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