Difference Equations

Part 1: Examples and Explorations

Answer questions 1 and 2 here.

Enter your formula for (Subscript[y, k + 1]) in place of ⬚⬚⬚ below.  The following commands are set up to generate 100 terms of the sequence from your formula and to print those terms with their term numbers.  You may adjust the number of terms as needed.

number = 100;
y[0] = 20000;

Do[                 y[
            k + 1] = ⬚⬚⬚,                       {k,  0,  
          number}];

ylist = Table[y[k], {k, 0, number}];
count = Table[k,  {k, 0, number}];
ypts = 
      Table[{count[[k]],  ylist[[k]]},  {k,  1,  
          number + 1}]

Answer Question 3 here.

The following command will plot the sequence.

ListPlot[ypts,  PlotStyle -> PointSize[ .02]];

Answer Question 5 here.

For Example 2,we repeat the commands for generating and plotting the sequence,but with the Samuelson formula instead.

number = 30;
a = 0.9
 b = 0.5
y[0] = 1
 y[1] = 1.1
Do[ 
        y[k + 2] = a*(1 + b)*y[k + 1] - a*b*y[k] + 1,  {k,  0,  
          number}];
ylist = Table[y[k], {k, 0, number}];
count = Table[k,  {k, 0, number}];
ypts = 
        Table[{count[[k]],  ylist[[k]]},  {k,  1,  
            number + 1}];
ListPlot[ypts,  PlotStyle -> PointSize[ .02]];

Answer the remaining questions in Part 1 here.

Part 2: First-Order Linear Difference Equations

Answer the questions in Part 2 here.You may set up and do any calculations necessary for answering these questions.

Part 3: Second-Order Equations

Answer the questions in Part 3 here.You may set up and do any calculations necessary for answering these questions.

Part 4: The General Case

Answer Questions 1-4 in Part 4 here.You may set up and do any calculations necessary for answering these questions.

For Question 5, you will need the following definition, which sets up a Vandermonde matrix.

R = 
        Table[r[i]^j,  {j,  0,  2},  {i,  1,  
            3}];
R // MatrixForm
Factor[Det[R]]

Answer Question 5 here.

Part 5: A Third-Order Equation

Answer the questions in Part 5 here.You may set up and do any calculations necessary for answering these questions.

Part 6: Summary

Answer the summary questions here.