The Fibonacci sequence F1; F2;::: is defined byF1 D 1; F2 D 1; and Fn D Fn2 C Fn1 for n
Chapter 5, Problem 16(choose chapter or problem)
The Fibonacci sequence F1; F2;::: is defined byF1 D 1; F2 D 1; and Fn D Fn2 C Fn1 for n 3:Define T 2 L.R2/ by T .x; y/ D .y; x C y/.(a) Show that T n.0; 1/ D .Fn; FnC1/ for each positive integer n.(b) Find the eigenvalues of T.(c) Find a basis of R2 consisting of eigenvectors of T.(d) Use the solution to part (c) to compute T n.0; 1/. Conclude thatFn D 1p51 C p52n1 p52n for each positive integer n.(e) Use part (d) to conclude that for each positive integer n, theFibonacci number Fn is the integer that is closest to1p51 C p52n
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