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Fibonacci Sequences, Convergence and Limits

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship


where F1=1, F2=1.

(b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1.

(c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fibonacci relationship. Is the Fibonnaci sequence that you found in (a) given by this formula for some A?

(d) Notice that the quadratic equation satisfied by r above has two roots. Let them be r1 and r2, and show that Ar1n+Br2n satisfies the Fibonnaci relationship, for any choice of constants A and B.

(e) Use the observation in (d) to find a formula for the nth term of the Fibonnaci sequence, and prove that it works.

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Fibonacci Sequences, Convergence and Limits are investigated. The solution is detailed and well presented.