A famous sequence {fn}, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200, is defined by the recursion formula
f1 = f2 = 1, fn+2 = fn+1 + fn
(a) Find .13 through f10.
(b) Let ( = ½(1 + (5) ( 1.618034. The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that

Check that this gives the right result for n = 1 and n = 2. The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that

(c) Using the limit just proved, show that 4 satisfies the equation x2 - x - 1 = 0. Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are ( and -1/(, two numbers that occur in the explicit formula for fn?

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fn 2 V5 lim fn+1/fn= φ.