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1. (Linear Derence Equations) The famous sequence E, of Fibonacci numbers is described by the recurrence relationship Fn : Fn_1 + F -2 with F0 2 0 and F1 : 1. The famous sequence of Lucas numbers is described by the same recurrence relationship, but with the different initial conditions F0 : 2 and F1 : 1 (a) Explain how these sequences can be viewed from the perspective of homogeneous linear difference equations. (b) Using the polynomial-rooting methods we described in lecture for nding solutions to homogeneous linear difference equations, derive a general expression for the family of solutions to the recurrence relationship Fn : Fn_1 + Fn_2. Your solution should have the form Fn 2 23:1 AgAn, where the Ag coefcients can be chosen arbitrarily but the Ag parameters are specic numbers. (c) How should the Ag parameters be chosen in your answer to (b) if you want E, to reproduce the Fibonacci sequence for n 2 0? (d) How should the Ag parameters be chosen in your answer to (b) if you want E, to reproduce the Lucas sequence for n 2 0? (e) Now consider a system with input :r[n] and output E, that is governed by E, = n_1 + Fn_2 + :r[n]. Assume that E, = 0 and :r[n] : 0 for Vn