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The fundamental relation Fn = Fn+2
- Fn+1 can also be used to define Fn
when n is a negative integer. Letting n = -1 in
this formula gives us F-1 = F1 - F0
= 1 - 0 = 1. Similarly, one finds that F-2 = F0
- F-1 = 0 - 1 = -1 and F-3 =
F-1 - F-2 = 1 - (-1) = 2. In this way one can
obtain Fn for any negative integer n. Some of
the values of Fn for negative integers n are shown
in the following table:
Perhaps the greatest investigator of properties of the Fibonacci and related sequences was François Edouard Anatole Lucas (1842-1891). A sequence related to the Fn bears his name. The Lucas sequence, 2, 1, 3, 4, 7, 11, 18, 29, 47, ..., is defined by
Some of the many relations involving the Fn and the Ln are suggested by the problems below. These are only a very small fraction of the large number of known properties of the Fibonacci and Lucas numbers. In fact, there is a mathematical journal, The Fibonacci Quarterly, devoted to them and to related material.
Problems for Chapter 2
1. For the Fibonacci numbers Fn show that:
(a) F3 = 2F1 + F0.
(b) F4 = 2F2 + F1.
(c) F5 = 2F3 + F2.
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Tuesday, February 17, 1998