The famous Fibonacci Sequence is defined by f0=fl = l, fn+2 = fn + fn+1, n = 0,

Question

The famous Fibonacci Sequence is defined by f0=fl = l, fn+2 = fn + fn+1, n = 0, 1.2, ... This problem is a guided tour towards finding explicit expressions for the terms of this sequence. To get started, compute the next few terms: We begin by looking at the general solution of the (homogeneous linear) difference equation and then finding the one that satisfies the starting conditions f0 = f1 = l. The creative step is to make an inspired guess of the form of some solutions of the difference equation: fn = rn for some constant r. Substituting in the difference equation and reorganizing gives Therefore fn = rn is a solution if r is a root of the quadratic equation r2 - r - 1 = 0. The smaller of those roots is

Explanation / Answer

f2 = 2

f3 = 3

f4 = 5

f5 = 8

f6 = 13

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