Abstract
The Induction Principle is of great importance in discrete mathematics: Number Theory, Graph Theory, Enumerative Combinatorics, Combinatorial Geometry, and other subjects. Usually one proves the validity of a relationship f(n) = g(n) if one has a guess from small values of n. Then one checks that f(n) = g(1), and, by making the assumption f(n) = g(n) for some n, one proves that also f(n + 1) = g(n + 1). From this one concludes by the Induction Principle that f(n) = g(n) for all n ɛ N.There are many variations of this principle. The relationship f(n) = g(n) is valid for 0 already, or, starting from some n0 > 1. The inductive assumption is often f(k) = g(k) for all k < n, and, from this assumption, one proves the validity of f(n) = g(n). We assume familiarity with all this and apply induction in unusual circumstances to make nontrivial proofs. We refer to Polya [22] to [24] for excellent treatment of induction for beginners. The reader can acquire practice by proving some of the innumerable formulas for the Fibonacci sequence defined by F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn, n ≥ 0. We state some of these.
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© 1998 Springer-Verlag New York, Inc.
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(1998). The Induction Principle. In: Problem-Solving Strategies. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/0-387-22641-9_8
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DOI: https://doi.org/10.1007/0-387-22641-9_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98219-9
Online ISBN: 978-0-387-22641-5
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