Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
WorkSheet 9
2
1
of the people outside Indiana move in, and 10
of the people inside Indiana
1) Each year 10
move out. If in year 0, there are x0 inside Indiana and y0 people outside Indiana, what
happens to the population long term? Notice that we are assuming that the total population
x0 + y0 does not change.
2) The Fibonacci sequence: let F0 = 1; F1 = 1; and Ã
for n ¸ !0; Fn+2 = Fn+1 + Fn : Find a
Fn+1
formula for Fn : Hint: if you de…ne the vector un =
; …nd the matrix A so that
Fn
un+1 = Aun: Now be creative in …nding a related way to write un+1 (this will involve powers
of A):