Chapter 10

Solving Linear Recurrence Relations: The Binet Form for F_{n}

Up to this point, if we wanted to know a specific Fibonacci number—for instance, F_{37}—we had to start with F_{0} = 0, F_{1} = 1, and use the recurrence relation F_{n} = F_{n−1} + F_{n−2}, n ≥ 2, to compute

Now our objective is to somehow determine F_{37}, but without performing any of these 36 calculations. To do so, we want to derive an explicit formula for the general term F_{n} in terms of n—not in terms of previous values in the sequence of Fibonacci numbers. We shall accomplish this by introducing the following idea:

Definition 10.1: For real constants C_{0}, C_{1}, C_{2}, . . ., C_{k}, with C_{0} ≠ 0 and C_{k} ≠ 0, an expression of the form

where n ≥ k, is called a kth-order linear homogeneous recurrence relation with 53 constant coefficients. When the right side of this recurrence relation is not 0, then the relation is referred to as nonhomogeneous.

We shall focus our concern on the case where k = 2, C_{0} = 1, and C_{2} ≠ 0. For example,

or

is a second-order linear homogeneous recurrence relation with constant coefficients.

To solve such a recurrence ...