One application of diagonalization is finding an explicit form of a recursively-defined sequence-a process is referred to as "solving" the recurrence relation. For example, the famous Fibonacci sequence is defined recursively by fo=0, fi = 1, and fm+1 = fm-1+fn for n 21. That is, each term is the sum of the previous two terms. (We could just as easily have written fn = fn-2+ fn-1 for n 2 2.) For example, using n = 1 we get f2 = fo+ fi = 0+1=1. Then using n=2 we get f3= f1 + f2=1+1=2. Then using n = 3 we get f = f2+f3=1+2=3. Then using n = 4 we get fs = f3+f4=2+3=5. Then using n = 5 we get fe= fa+fs=3+5=8 and so on. The Fibonacci sequence therefore begins 0,1,1,2,3,5,8,13, 21, 34, 55, 89, 144,... In this assignment we will find an explicit form of the Fibonacci sequence. 1. Consider the matrix A A³ = [B] Calculate the following powers of A by hand (show some work): fa+l fn 2. Based on our examples, we can reasonably conjecture that A" -=[₁ ₁] for prove our conjecture we would use mathematical induction. The key step is calculating A = AA-1 BLAH Complete this calculation and simplify your answer. 3. Before we continue our analysis of the Fibonacci sequence, note that we could have constructed A from the recursion as follows: + []-[-]-[-]--B-CRIA] Just for practice (having nothing to do with Fibonacci's), construct the matrix B such that [K][S] for the sequence b,, satisfying the recursion b136-1-26, for n 2 1. tn 21. To + + 4. Returning to our analysis of the Fibonacci sequence, we would like to find an explicit form of the sequence. Calculate the characteristic polynomial for A and solve the characteristic equation to find the eigenvalues of A. Hint: quadratic formula! Show calculations by hand.