Sequences -- Cobweb Diagrams -- Stretch Your Understanding

1. A linear dynamical system of the form

   p(n + 1) = m p(n) + b

   where m and b are both constants and m is not 1 has exactly one equilibrium point found by solving the equation

    
          p = mp + b
     p - mp = b
   p(1 - m) = b 
          p = b / (1 - m)
   
   

   This means that if p(1) = b / (1 - m) then all the remaining terms will have the same value.

   We have seen many examples of discrete dynamical systems in which the sequence appears to be pulled in or attracted to an equilibrium. By looking at cobweb diagrams for this particular family of models see if you can determine when the sequence is pulled in to the equilibrium point. See if you can formulate a theorem that answers this question.