Math 2602 B1 , B2                                                                                            Homework # 10

                                              Due Thursday , November 13 , 2003

   Let { a_n } and { b_n } be the sequences defined by a₀ = 1 and b₀ = 0 , and a_n = 2a_n-1 +  b_n-1 ,
   b_n = a_n + 2b_n-1 , for 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 = ( 1 , 0  )^T as  ru + sv , where u and v are independent eigenvectors of
          the matrix  A .


  (iii)  Find a_n and b_n .



  (iv)  Does  a_n have a limit?   Is  a_n = o( b_n ) ?