Math A1 - A3                                                                                             Homework  # 9
                                            Due Thursday, April 1, 2004


Let { a_n } and { b_n } be the sequences defined by a₀ = 2 and b₀ = 3 , and a_n = 1/2 a_n-1 + 1/2 b_n-1 ,
b_n = 1/3 a_n-1 + 2/3 b_n-1 .

i)  Find a matrix  A such that  A ( a_n-1 , b_n-1 )^T = ( a_n , b_n )^T .

ii)  Write ( a₀ , b₀ )^T = ( 2 , 3 )^T as  ru + sv , where  u  and  v  are independent eigenvectors of the
      matrix  A .

iii)  Find  a_n and b_n .

iv)  Find the limits of  a_n and  b_n as n increases .