Direct Proof…An Alternative To
Proof By Induction
The main purpose of this paper is to introduce the reader to an alternative to proof by induction. While this document is not meant to replace existing material on the subject, it does offer new, innovative concepts. My document will introduce the reader to a method of defining all Fibonacci numbers with a single parameter. The result of this allows formula to be proven directly. Proofs are therefore easily understood. As a classroom tool for teaching Fibonacci numbers and Lucas numbers, this paper is a necessity.
Beyond that, the same principles are extended to the universe of all Fibonacci-like sequences. The reader will be able to explore relationships of other similar sequences.
Two formulas used in the development of this paper were actually created and copyrighted in another of my manuscripts, In Search of Pi. When given F(n), F(n + 1) can be computed exactly and directly as follows:
F2n = (F2n−1 + (5 * F22n−1 − 4)½) / 2
F2n+1 = (F2n + (5 * F22n + 4)½) / 2
About Edgar M. Alexander
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Published November 29, 2011
Professional & Technical, Science & Math, Education & Reference.