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Transcript

Fibonacci Sequence Worksheet In this worksheet, we will use linear algebra to study the Fibonacci sequence. Let us first recall the definition of the Fibonacci sequence. Let F0 = 1, F1 = 1, and for n 2, let Fn = Fn 1 + Fn 2 . We call Fn the n-th Fibonacci number. Thus, the Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, . . . Notice that our definition of Fn is recursive, meaning that to compute Fn you must know previous values in the sequence. Among other things, we will attempt to rectify this situation. F 1 1 (a) Let us package the Fibonacci numbers into vectors xn = n 1 . So, for example, x1 = , x2 = , and Fn 1 2 5 x5 = . Use the recursive definition of Fn to find a 2 ⇥ 2 matrix A such that Axn = xn+1 . 8 (b) A has two real, distinct eigenvalues 1 > 2. Determine (c) Find bases for the 1 dimensional eigenspaces E 1 1 and 2. and E 2 . (d) Let v1 and v2 be the bases for E 1 and E 2 , respectively. Then B = {v1 , v2 } is a basis for R2 . Use this basis to diagonalize A; that is, write A = SDS 1 for some matrix S and some diagonal matrix D. (Hint: All the heavy lifting has been done in the previous parts of this problem.) (e) Use part (d) and the fact that An x1 = xn+1 to find a closed form expression for Fn . That is, give a formula for Fn which depends only on n. Notice that this formula involves irrational numbers, but from the recursive definition of Fn , we see that it will always produce integers! (f) Determine limn!1 Fn+1 Fn . (g) Go to Wikipedia and read about the “Golden Ratio”. Marvel at the work you’ve done and the connections you’ve just found. 1