Exercise 11.  Solve the difference equation      with the initial condition   .  

Use recursion (and mathematical induction) to find the solution.

That is, compute  ,  ,  ,  then find the general term.  

Solution 11.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Use the recursive formula    to find the solution with initial condition  .  The first few terms look like

                    ,  

                    ,  

                    .  
        
                    Assume that    has the form
        
                      
        
                    then the next step is
        
                    


Therefore, we have established the formula by mathematical induction.  

We are done.   

We can let Mathematica verify a few of the sums.

          The Maple commands are similar  

  

                                                            


  

                                                            


  

                                                            

We are really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

          The Maple command is similar  

  

                                                            

We are really really done.   

Remark.  If we observe that    then the line    can be written as

                    .

Now use and combine terms to get
        
                    ,

which is the convolution form of the solution.

We are really really really done.   

Aside.  We can explore the situation when   

                                                                                                                        The sequence   .   

Aside.  We can explore the situation when   

                                                                                                                        The sequence   .   

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell