7.2.2 Fibonacci Numbers Denote the kth Fibonacci number as F. These numbers are defined by the rule that the kth number is equal to the sum of the two preceding ones with the initial numbers being Fo = 0 and F1 = 1. They satisfy the following difference equation: Fr = Fk-1+ Fk-2. (7.36) This is a second-order, linear difference equation with constant coefficients and can be easily solved. Setting Fk = rk gives the characteristic equation p² – r – 1= 0, (7.37) whose solutions are 1+ v5 ri = 1– V5 r2 = (7.38) 2 Consequently, the general solution of equation (7.36) is k 1+V5 1- V5 F = C1 + C2 (7.39) The arbitrary constants C1 and C2 may be found by imposing the initial conditions Fo = 0 and F¡ = 1. Doing this gives C1 + C2 = 0, (1+ v5)C1 + (1 – V5)C2 = 2, (7.40) which, when solved for C1 and C2, gives 1 C1 = -C2 = (7.41) V5 APPLICATIONS 219 Putting these values in equation (7.39) gives, for the kth Fibonacci number, the result + V5 V5 1 F = V5 (7.42) 2 In the remainder of this section, we will examine some of the properties of these numbers. First, consider the limit Fr+1 lim (7.43) k¬0 F From equation (7.42), we find Fr+1 FR 1 - (1 – V5/(1+ v5)]*+1\ 1– (1 – v5)/(1+ v5)]* S* 1+ v5 (7.44) - Since |1- V5 < 1, |1+ V5 (7.45) then 1+ v5 Fr+1 lim k0o F. = 1.618033988. ... (7.46) This number, called the golden mean, represents the ratio of the sides of a rectangle that is supposed to be “most pleasing" to the eye. Consider now the values of the Fibonacci numbers for the index k being a negative integer. The most direct way of obtaining these values is to substitute -k for k in equation (7.42). Doing this gives