Assignment:
Qa) Find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship
Fn=Fn-1+Fn-2
where F1=1, F2=1.
Qb) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1.
Qc) 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?
Qd) 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.
Qe) Use the observation in (d) to find a formula for the nth term of the Fibonnaci sequence, and prove that it works.
Provide complete and step by step solution for the question and show calculations and use formulas.