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Topic: Proof of mathematical induction

	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.
		Indeed, the validity of mathematical induction is logically equivalent to the well-ordering principle.
		Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics.
	en.wikipedia.org /wiki/Proof_of_mathematical_induction (2665 words)

	Teaching Math: Grades 9-12: Reasoning and Proof
		On the topic of induction and deduction, there is one other term to introduce and define, proof by "mathematical induction" which constitutes a special case.
		Frequently a row of dominoes or a ladder is invoked as a handy metaphor for mathematical induction.
		Mathematical induction is a technique that students should be familiar with by the end of high school student.
	www.learner.org /channel/courses/teachingmath/grades9_12/session_04/section_03_e.html (446 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.
		Although this proof seems to write itself this is certainly not always the case of induction proofs, but it does give an idea of how useful a tool that induction is.
		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)

	Induction and mathematical induction (Site not responding. Last check: 2007-11-01)
		Induction is the process of discovering general laws by the observation and combination of particular instances.
		Mathematical induction is used in mathematics alone to prove theorems of a certain kind.
		Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science.
	webpages.charter.net /rivendell/heuristics/induction_and_mathematical_induction.htm (1635 words)

	Peter Suber, "Mathematical Induction"
		Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class.
		The induction step is the proof of a conditional statement, namely, "if the theorem is true of the ancestor case, then it is true of the descendant cases." The if-clause of this conditional statement, asserting that the theorem is true of the ancestor case, is called the induction hypothesis.
		It is assumed for the sake of a conditional proof; we don't have to prove it.
	www.earlham.edu /~peters/courses/logsys/math-ind.htm (1191 words)

	Making Mathematics: Mathematics Tools: Mathematical Induction (Site not responding. Last check: 2007-11-01)
		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)

	The count of regions of a circle cut by chords (Site not responding. Last check: 2007-11-01)
		However, the process that has been used is not mathematical induction at all, but rather a form of inductive reasoning which is very common and very useful, while being also very dangerous.
		Mathematical induction is usually reserved for study by a select few at the high-school level and may even then be poorly presented and even more poorly understood.
		A rigorous presentation of proof by mathematical induction requires that at least some formal development of the natural numbers has been undertaken, at least to the extent of introduction to something like the Peano postulates.
	www.umpi.maine.edu /~kimball/regionct.htm (779 words)

	Mathematical Induction (Site not responding. Last check: 2007-11-01)
		Mathematical Induction works like this: Suppose you want to prove a theorem in the form "For all integers n greater than equal to a, P(n) is true".
		Math induction is of no use for deriving formulas.
		But it is a good way to prove the validity of a formula that you might think is true.
	zimmer.csufresno.edu /~larryc/proofs/proofs.mathinduction.html (435 words)

	Mathematical deduction and mathematical induction (Site not responding. Last check: 2007-11-01)
		Mathematical induction is a particular type of mathematical argument.
		Mathematical induction is a way show how to "keep going".
		Proof by mathematical induction proceeds in two steps.
	mathcentral.uregina.ca /qq/database/QQ.09.99/pax1.html (316 words)

	indexcourse
		An alternative to a direct proof is proof by mathematical induction.
		Use proof by mathematical induction to justify any rules suggested by the investigations.
		Proofs by induction - investigations 1 to 4
	www.teachers.ash.org.au /mikemath/numseqinduct (310 words)

	Lecture 11 for Math 232
		Mathematical induction is presented in section 3.2 of Rosen.
		Beginners at induction proofs find themselves saying that "P(n) is equal to some number", which makes no sense.
		The recipe for an induction proof takes a special form when the statement P(n) is of the form "S(n)=T(n)", where S(n) and T(n) are functions yielding numbers.
	www.mtholyoke.edu /courses/barring/232/lecture/11.htm (882 words)

	Proof by Mathematical Induction
		Proof by Mathematical Induction is a method of proof which Computer Scientists encounter on a regular basis.
		Intuitively, this method of proof uses the validity of one instance of an algorithm to prove the validity of another instance.
		After such a proof is constructed we are left with the statement - IF there is an instance of the algorithms that is true THEN there is this second instance which we have proved to be true.
	www.neiu.edu /~css/archive/induction.html (648 words)

	Algebra Finite Mathematics Calculus 25 Inductive Proof.Eg
		It is a method of proof, which relied on a sequence of assertions, each one of which implied that it successor == the next one in the sequence.
		Mathematical induction is used in the first section to confirm or justify the addition formulas given earlier for geometric and arithmetic sums, and to say a little more.
		Mathematical induction is employed to justify the earlier formulas for the values of these sums.
	whyslopes.com /etc/ThreeSkillsForAlgebra/ch25.html (478 words)

	Proof by Mathematical Induction
		Mathematical Induction is often used in situations where the problem can be stated as a list of examples.
		Therefore we have proved by Mathematical Induction that the statement ‘that the sum of a number of odd numbers can be immediately found by taking the square of how many odd numbers there are’ is correct.
		Step 3: Therefore by the Principle of Mathematical Induction P(n) is true for all values of n in the Natural Numbers.
	indigo.ie /~hallinan/mathPgs/Higher_L_Cert/Proof_by_Mathematical_Induction.htm (918 words)

	Computational Recursion and Mathematical Induction (Site not responding. Last check: 2007-11-01)
		A recursive process invokes a version of itself as a subprocess during execution; a recursive procedure contains a version of itself as a subprocedure in its definition; in a proof by mathematical induction, one version of the theorem is used to derive another version.
		Both the mathematical and the computational processes are based on the same recursive idea: If the given number is prime, output that number; else, decompose the given number into a product of two smaller numbers and output the product of their prime decompositions.
		In induction, it is the validity of your proof, a very mysterious entity that even adult students find hard to evaluate.
	cse.proj.ac.il /recursion/Induction_Recursion.htm (2639 words)

	List of mathematical proofs - Wikipedia, the free encyclopedia
		2 Articles devoted to theorems of which a (sketch of a) proof is given
		Proof that the sum of the reciprocals of the primes diverges
		Articles devoted to theorems of which a (sketch of a) proof is given
	en.wikipedia.org /wiki/List_of_mathematical_proofs (195 words)

	Research Interests
		Proof Planning: We have pioneered the use of proof planning as a global strategy for guiding reasoning.
		Reasoning using induction is a key aspect of proof in higher-order logics, yet is inherently complex.
		In particular, by viewing a proof as being distributed amongst autonomous agents, we may be able to combine the benefits of parallel search in large problem spaces, with agent-based aspects in complex problem spaces.
	www-users.cs.york.ac.uk /~frisch/AutoReason/98/statements.html (1332 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 +...
	www.math.sc.edu /~sumner/numbertheory/induction/Induction.html (1878 words)

	seminars2002.html
		With a grant from the Stuart Foundation to the mathematics department at the University of California at Berkeley and to the University of Chicago School Mathematics Project, mathematics materials are being developed for high school teachers in an attempt to conceptualize this field of study.
		Implicit in the design of the experiment were two assumptions, that proof by mathematical induction would arise (a) as the means for solving a class of problems and (b) as a result of the students’ mathematization of the "and so on" argument.
		The Mid-Atlantic Center for Mathematics Teaching and Learning, a collaboration among the mathematics and mathematics education faculty at Penn State, University of Maryland, and University of Delaware, is the first Center for Mathematics Teaching and Learning funded by the National Science Foundation.
	www.math.uic.edu /~imse/IMSE/seminars2002.html (1283 words)

	MAT 140: Discrete Mathematics I
		That is, suppose...,” and continue this sentence by carefully writing the negation of the statement to be proved.
		Mathematical induction is used to prove properties about integers.
		A proof by mathematical induction consists of two parts.
	condor.depaul.edu /~sepp/MAT140/ProofTipsW04.htm (336 words)

	Math 2305 Reading Assignments
		State mathematical induction as a rule of inference.
		After reading a few of the examples of proof by mathematical induction, describe a common element in the statements (not the proofs) that were proven.
		I am interested in knowing if you were able to surmise when to use the theorem of mathematical induction.
	cms.dt.uh.edu /faculty/delavinae/sm02/R3_2.htm (75 words)

	PROOF IN MATHEMATICS: AN INTRODUCTION
		This is a small (98 page) textbook designed to teach mathematics and computer science students the basics of how to read and construct proofs.
		True, creating research-level proofs does require talent; but reading and understanding the proof that the square of an even number is even is within the capacity of most mortals.
		Proof in Mathematics: an Introduction takes a straightforward, no nonsense approach to explaining the core technique of mathematics.
	web.maths.unsw.edu.au /~jim/proofs.html (368 words)

	Mathematical induction - Topics in precalculus
		The method of proof follows from the following, which is called the principle of mathematical induction.
		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 (988 words)

	[No title]
		Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory.
		Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction.
		Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars.
	aduni.org /courses/discrete/courseware/syllabus/General_Description.doc (400 words)

	7.4 - Mathematical Induction
		One of these methods is the principle of mathematical induction.
		The conclusion is found by saying "Therefore, by the principle of mathematical induction" and restating the original claim.
		After you have your pattern, then you can use mathematical induction to prove the conjecture is correct.
	www.richland.edu /james/lecture/m116/sequences/induction.html (1919 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 (363 words)

	Section (iii) Mathematical Induction and the Triangle Inequality: Cultivating More Fruitful Uses of Intuition and ...
		Structure: In the following the tutor presents a 'back-to-basics' proof of the Base Statement of the Mathematical Induction for the triangle inequality as opposed to the geometric proof given in the lectures.
		(b) the tutor's pragmatic response to where the proof by mathematical induction of the triangle inequality should start from (the triangle inequality 'is a basic assertion for two.
		The competence demonstrated in Andrew's elaboration on the eight possibilities in the proof of the triangle inequality for two numbers is remarkable: not only does he point out to the tutor the possibility of reducing the number of cases to be written down but also helps the tutor realise what this number is.
	www.uea.ac.uk /~m011/thesis/chapter6/6iii.htm (797 words)

	[No title]
		Performs tactical proof by mathematical induction on the natural numbers.
		P[n], where n has type num, to two subgoals corresponding to the base and step cases in a proof by mathematical induction on n.
		The induction hypothesis appears among the assumptions of the subgoal for the step case.
	www.cl.cam.ac.uk /users/jrh/hol-light/HTML/INDUCT_TAC.html (252 words)

	Welcome to the Proof Project (Site not responding. Last check: 2007-11-01)
		Barki, R. and Tirosh, D. and Tsamir, P. (2004) Is it a mathematical proof or not? Elementary school teachers’ responses. Proceedings of the 28th conference of the International group for the Psychology of Mathematics Education.
		Contreras, J. (1998). Effects of instruction on students’ construction of proofs: prospective elementary and secondary teachers and the case of the angle sum in the a triangle theorem.
		DeVilliers, M.D. Rethinking proof with the Geometer’s Sketchpad.
	www.theproofproject.org /bibliography (8656 words)

	Induction Based on Rippling and Proof Planning. Mini-Course (Site not responding. Last check: 2007-11-01)
		Mathematical Induction is a central technique in reasoning about programs and their properties, e.g., loops and recursion, recursively defined data-structures, and program termination.
		In this five hour seminar I will cover some of the central issues in automating proof by mathematical induction.
		In particular, formalisms for mathematical induction, techniques for selecting induction schemata and well-founded orders, rewriting in inductive theorem proving, and applications.
	www.brics.dk /NS/94/2 (108 words)

	FLoC'99 Workshop on Automation of Proof by Mathematical Induction: Cfp
		Proof by Mathematical Induction presents the Automated Deduction community with some very challenging research problems.
		Induction is one of the key techniques when dealing with abstract datatypes.
		The aim of this session is to address the adequacy of current automated techniques for inductive proof within the context of higher-order logics.
	www.dfki.de /floc-ws13 (607 words)

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