Exercise 18.  In the value of an annuity due model     

use the parameters   .

18 (a).  Use the z-transform and tables to find the solution.    Hint.  Get  .  

18 (b).  Use residues to find the solution.

Solution 18.

See text and/or instructor's solution manual.

Answer.   .  

Solution 18 (a).

        Substitute      into   ,  

and get the difference equation  

                    .  

Take the z-transform of both sides and use the assumed initial condition  :  

                    ,  

                    .  
        
Solve for    and get  

                    ,  

then use Table 9.1 to find the inverse z-transform   

                      

Remark.  The details for the partial fraction expansion are at the bottom of the page.

We are done.   

Solution 18 (b).

        Using residues we get

                      

                    and

                      
                    
Thus

                    

Therefore,

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Take the z-transform of both sides.

Find the inverse z-transform.

          The Maple commands are similar  

  

                                                            


  

                                                            


  

                                                            


  

                                                            

Aside.  We can use Mathematica's InverseZTransform subroutine.

          The Maple command is similar  

  

                                                            

We are really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

          The Maple command is similar  

  

                                                            

Aside.  The solution can be written as  

                    .  

We are really really done.   

Aside.  We can graph the solution.

                                                                                                                        The sequence   .   

We are really really really done.   

The Details for the Partial Fractions.   

Aside.  How can we expand      into the proper partial fractions?

It is natural to try the command:

But this is not the desired form for using Table 9.1 of z-transforms.

Method (i).   Use the following algebra steps  

                      

Method (ii).   Find the linear combination of   ,  

                    .  

Equate the numerators   ,  

and solve the linear system  

                     

and get   .   

Therefore, the desired form is  

                    .  

Aside.   The Mathematica commands for Method (ii)  are

Method (iii).  (For distinct real roots)   First make the substitution      in      and get  

                    .  

Then use the standard procedure for expanding in partial fractions   

                    .  

Then make the substitution      in      and get  

                      


Therefore, the desired form is  

                    .  

Aside.   The Mathematica commands for Method (iii)  are

Now use the substitution

                    .     
                    
Therefore, the desired form is  

                    .  

This solution is complements of the authors.

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