Math 110 Homework Assignment 18 due date: Mar. 3, 2017 1. The Fibonacci Numbers are the numbers defined by F0 = 0, F1 = 1, and Fn = Fn1 +Fn2 for n 2. (So for instance F2 = F1 +F0 = 1+0 = 1, F3 = F2 +F1 = 1+1 = 2, and F4 = F3 + F2 = 2 + 1 = 3, etc.) \u0014 \u0015 1 1 (a) Suppose that we set w ~ n = (Fn , Fn1 ) for n 1, and A = . 1 0 Show that the recursion relation above means that w ~ n+1 = Aw ~ n. (b) Use the eigenvectors of A to find a formula for Ak w ~ 1 for any k 0. (c) Use the answer from (b) to find a formula for the n-th Fibonacci number Fn . (d) The Lucas Numbers are the numbers defined by L0 = 2, L1 = 1 and Ln = Ln1 + Ln2 for n 2. Find a formula for the n-th Lucas number. 2. Let A be the matrix A= \u0014 19 14 21 16 \u0015 . Find a formula for the entries of Ak (for k 1). As a check, compute A2 and A3 and to see if they match your formulas. Hint: The first column of any 2 2 matrix is \"where (1, 0) gets sent\