Page 1 of 18

After completing this tutorial you will be able to:

Enter sequences which are recursively or explicitly defined.

Set the WINDOW for each sequence.

Generate a table or graph for each sequence.

You will do this using some pollution and pollution control models.
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

Page 2 of 18

The TI-82 sequence keys which we will use extensively in this tutorial are outlined in blue on the screen shown at the right.

Typing Convention

Functions accessed by pressing a primary key will be indicated by “press ‘key name’”, for example press GRAPH.

Functions accessed by pressing the 2nd key first will be indicated by “press 2nd, ‘primary key name’”. For example entering U n–1
would be indicated by “press 2nd, 7”

Common combination symbols for this tutorial
 u n–1
= 2nd, 7

 v n–1
= 2nd, 8 n = 2nd, 9

TblSet = 2nd, WINDOW

TABLE = 2nd, GRAPH
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

Page 3 of 18

The TI-82 must be put in sequence mode to work with sequence definitions. To do this:

Press MODE to display the MODE settings.

Use the cursor keys to highlight Seq and then press
ENTER to select Seq which is the last element in the 4 th row.

You may also wish to select Dot (2 nd element in the 5 th row) to show discrete values when graphing.

These setting are shown in the top screen at the right.

For this tutorial the WINDOW FORMAT settings should be as shown in the bottom screen at the right. To do this:

Press WINDOW and then use the right cursor key to highlight FORMAT.

Use the cursor keys and the ENTER key to make your settings conform to those shown.

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Page 4 of 18

Initially there are 500 pounds of contaminant in a lake. 10% of the contaminant is washed away each year. However, a manufacturing concern annually dumps 100 pounds of the contaminant into a river that feeds the lake. Will the level of contamination continually increase or will it stabilize at some level?

Let u n
represent the amount of contaminant in the lake after n-years and the initial amount u
0
=
500.

After 1-year 90% of the 500 lbs would remain at which time an additional 100 lbs would be dumped into the lake. Therefore, u
1
= 0.90 u
0
+ 100.

This suggests that u n
= 0.90 u n–1
+ 100; u
0
= 500, n

1 is the recursive equation for the amount of contaminant in the lake after n-years.

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Page 5 of 18

Press Y= to enter the sequence equations list.

Two functions, U n
and V n
, defined either recursively or explicitly may be entered.

Clear any previously entered functions by using the CLEAR key and the cursor keys.

Enter U n
as 0.9 U n–1
+ 100 by pressing:
. , 9, 2nd, 7, +, 1, 0, 0
Leave V n
undefined.

The resulting screen is shown at the right.
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

Page 6 of 18

Press WINDOW to go to the WINDOW setup screen. The complete WINDOW setup screen in shown in the first two screens at the right (which have been placed to appear as one).

General rules for setting WINDOW values

The starting value of your sequence(s) will define
U n Start, V n Start and n Start, u
0
= 500 for our example.
 n Min = n Start unless you want the graph to start at a sequence value different from the sequence value at n Start.
 n Max = a value of your choosing

Xmin, Xmax = n Min, n Max unless you want the graphing window to be different from the sequence range.

Ymin and Ymax are values of your choosing and will depend upon the minimum and maximum values of the sequence.

Xscl and Yscl should be chosen so that the X and Y axes do not become too thick and interfere with your view of the graph or become distracting to look at. For example, Xscl = (Xmax –
Xmin)/10 is a setting that works well for me in most instances.

Use ENTER and the keypad to make your WINDOW setting the same as those shown.

Press GRAPH to display a partial graph of the sequence. (Your graph will appear very similar to the graph shown on the Exercise 1 answer page.)

Exercise 1 : Use the TRACE key to help decide if the level of contamination continually increases or will stabilize at some level. You may wish to reset your WINDOW values and GRAPH several times to help you decide.

Click here to Goto Exercise 1 answers .

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Page 7 of 18

To generate a table of values for our recursive equation, u n
=
0.9 u n–1
+ 100, u
0
= 500, n

1:

Go to TblSet (2ND, WINDOW). Use the cursor keys and the ENTER key to make your TABLE SETUP conform to the screen shown at the right.

Press TABLE (2ND, GRAPH) to display a table of values for the function as shown in the bottom screen at the right.

Use the down cursor key to cursor through the values and see once again that the limiting value of the contaminant in the lake will be 1000 lbs. (This will take some time.)
 Exercise 2 : QUIT (2ND, MODE) to the home screen. Use the DRAW (2ND, PRGM) function 3:Horizontal and
ENTER to draw a horizontal line at 1000 to see a graphical representation of the limiting value for the contaminant.

Click here to Goto Exercise 2 answers
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

Page 8 of 18

The explicit solution for the recursive equation u n
= 0.9 u n–1
+ 100; u
0
= 500, n

1 can be shown to be u n
= –500 (0.9) n + 1000, n

0

We can easily see from the explicit equation that the limiting value of the contaminant must be
1000 lbs since the exponential portion will approach 0 as n gets large.

What may not be so easy to see is why the recursive definition has the explicit solution that it does. The solution comes from using techniques applied to first order linear difference equations.
You may consult either the ‘Algebra’ module or the ‘Dynamical Systems’ module for more information on how to write and solve first order linear difference equations.

To enter the explicit equation:

Press Y= to enter the sequence equations list.

Cursor down to V n
and enter the explicit equation: V n
= –
500(0.9)
 n + 1000
The keystrokes are:
(–), 5, 0, 0, (, ., 9, ), 2ND, 9, +, 1, 0, 0, 0

The resulting screen is shown at the top right.

Go to WINDOW and edit V n
Start from 0 to 500. The resulting screen is shown at the right..

Go to TblSet (2ND, WINDOW) and make sure that TblMin =
0.

Press TABLE (2ND, GRAPH) to see a table for both the recursively and explicitly defined functions. The tables should be identical as shown in the screen at the bottom right.

If you were to Press GRAPH at this time you would see just one graph. U n
and V n
may be selectively turned ON or OFF.
This will determine which graph or table is generated.

Exercise 3 : Turn off U n
. Graph V n
for 0

n

50.
Create a table of values for V n
by making TblMin = 0 and

Tbl =
1.
Create a second table for V n
with
TblMin = 106. (The TI-82 will grind for quite a while before you will see the table.)

Click here to Goto Exercise 3 answers .

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

Page 9 of 18

To help students understand how values for difference equations are generated I have them calculate sequence values by hand using the TI-82 as a simple calculator first.

I then explain that the ANS function (2ND, (–)) stores the result of the computation just performed and that we can easily use this idea to create a table of values for simple difference equations.

Let u n
= 0.9 u n–1
+ 100; u
0
= 500, n

1

The following will generate a table of values for u
0
, u
1
, u
2
, …
TI-82 home screen
500
Keystrokes
5, 0, 0
500 ENTER [500 = u o
]
.9* ANS + 100 ., 9, *, 2ND, (–), +, 1, 0, 0
550 ENTER [550 = u
1
]
.9* ANS + 100 2ND, ENTER [recalls previous computation]
595 ENTER [595 = u
2
]
 etc.

In fact this is the approach I use entirely in my classes as some students are still using the TI-81 which does not have sequence capabilities.

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

Page 10 of 18

Initially there are 500 pounds of contaminant in a lake. 10% of the contaminant is washed away each year. A manufacturing concern plans to annually dump 100 pounds of the contaminant into a river that feeds the lake. However, the EPA finds out about the dumping and requires the manufacturing concern to reduce the level of dumping by 25% per year after the initial dump of 100 pounds. Write a recursive equation which models this situation.

This situation seems to be similar to our first model which had the recursive definition u n
= 0.9 u n–1
+ 100; u
0
= 500, n

1 except that instead of adding a constant 100 pounds of contaminant per year we are adding smaller and smaller amounts.

These amounts would be:
100
100 – 0.25 (100) = 0.75 (100) = 75
75 – 0.25 (75) = 0.75 (75) 0.75 (0.75(100)) = (0.75) 2 * 100 etc.
 Looked at another way the additional amounts are {100, 75, 56.25, 42.1875, …}. This is an exponential sequence, a n,
generated by a n
= 100 (0.75) n – 1 ; n

1.

The recursive definition for our pollution model is therefore u n
= 0.9 u n–1
+ 100 (0.75) n – 1 ; u
0
= 500, n

1

Exercise 4 : Use a table of values to find u

Click here to Goto Exercise 4 answers .

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7
.
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

Page 11 of 18

The explicit solution, obtained again from solution techniques for first order linear difference equations, is u n
= 1166.6666 (0.9) n – 666.6666 (0.75) n , n

0

Exercise 5 : Will the level of contamination continually increase, stabilize at some higher level as in our last example or stabilize at some lower level? If it stabilizes at some lower level, what is the approximate maximum level of contamination? Use a graph and the explicit equation to determine the answer.

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
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

Page 12 of 18

The Fibonacci sequence u
1
= 1, u
2
= 1, u n
= u n–1
+ u n–2
, n

3 is an example of a second order difference equation and can be evaluated using the TI-82 by making use of both U n
and V n
as follows.

Press Y= and define U n
and V n
as:
U n
= V n–1
V n
= U n–1
+ V n–1

Press WINDOW and set the WINDOW as follows:
U n
Start = 1
V n
Start= 1 nStart=1 nMin=1 nMax=10
Xmin=1
Xmax=10
Xscl=1
Ymin=0
Ymax=60
Yscl=10
Press 2ND, WINDOW to get to TblSet and make TblMin =

Tbl=1
Press 2ND, GRAPH to get the table shown below. U n
will be the Fibonacci sequence. u U n
V n
1
2
1
1
1
2
3
4
5
6
7
2
3
5
8
13
3
5
8
13
21
Press Y=, turn-off V n
by cursoring to the = sign for V n
and then pressing ENTER and then Press GRAPH to see a graph of the sequence. [The graph was generated in Microsoft
Excel but is very similar to what you will see on the TI-82 screen.]
60
54
48
42
36
30
24
18
12
6
0
0 1 2 3 4 5 6 7 8 9 10

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

Page 13 of 18
Math
T E A C H E R
Link
Choose Edit, GoTo, Bookmark…,
* (from Enter bookmark name: pull down list), GoTo, Close to return to some point in the tutorial OR just exit the tutorial from here.

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
Press WINDOW to change the WINDOW settings to the values shown in the screen at the top right. Leave the rest of the settings as we had them. (If you have changed them, change them back to the original settings.)


Press GRAPH to get the graph shown at the bottom right.
An inspection of the functional values using TRACE suggests the contamination will stabilize at 1000 pounds as shown in the screen at the right.
Click here to Goto Exercise 1 page .
Page 14 of 18



Reset the WINDOW settings to show nMax and Xmax = 50.
Make sure you are on the home screen by pressing QUIT
(2ND, MODE). You may need to press CLEAR to make sure that you are at the beginning of a line.

Press 2ND, PRGM (DRAW) to enter the draw menu shown in the top screen at the right.

Cursor down to 3:Horizontal (highlighted in the screen at the right) and then press ENTER.

This will cause the last line of text on the home screen to read
Horizontal 

Complete the line to read
Horizontal 1000 

Press ENTER to get the graph screen shown at the right.
(You might wish to play around with different WINDOW settings at this time.)
Click here to Goto Exercise 2 page .
Page 15 of 18


To turn off U n
: go toY=, cursor to the = for U n
and press
ENTER so that the = sign is not highlighted (which turns U n off). See the screen at the right.

To graph V n
:

Go to WINDOW, if necessary, and make sure that nStart = nMin = Xmin = 0 and that nMax = Xmax = 50.
The Y scalings should still be ok.

Press GRAPH to get the graph of V n
shown in the 2 nd screen at the right.

To create the first TABLE:

Go to TblSet (2ND , GRAPH)

Edit TABLE SETUP so that TblMin=0 and

Tbl=1 as shown below.
TABLE SETUP
TblMin=0

Tbl=1
Indpnt: Auto Ask
Depend: Auto Ask

Press TABLE (2ND, GRAPH) to get the table shown in the 3 rd screen.

To get the second TABLE:

Go to TblSet (2ND , GRAPH)

Edit TABLE SETUP so that TblMin=106 and

Tbl=1 as shown below.
TABLE SETUP
TblMin=106

Tbl=1
Indpnt: Auto Ask
Depend: Auto Ask

Press Press TABLE (2ND, GRAPH) to get the table shown in the 4 th screen.
Click here to Goto Exercise 3 page
Page 16 of 18


To enter the recursive equation:

Press Y= to enter the sequence equations list.

Clear the previous equations using the CLEAR key and the cursor keys.

Enter the recursive equation for U n
. U n
= 0.9 u n–1
+ 100
(0.75) n – 1

 The keystrokes are ., 9, 2ND, 7, +, 1, 0, 0, (, ., 7, 5,
),

, (, 2ND, 9, –, 1, )

To create the TABLE:
Press 2 ND , WINDOW to get to TblSet

Set TlbMin = 1 or 7

Press 2ND, GRAPH to create the TABLE which will start at either n = 1 or n = 7 and show u
7
= 469.02.
Click here to Goto Exercise 4 page
Page 17 of 18

Page 18 of 18

To enter the explicit equation:

Press Y= to enter the sequence equations list.

Enter the explicit equation in V n
. V n
= 1166.6666 (0.9) n
– 666.6666 (0.75)

The keystrokes are: 1, 1, 6, 6, ., 6, 6, 6, 6, (, ., 9, ),
 n
, 2ND, 9, –, 6, 6, 6, ., 6, 6, 6, 6, (, ., 7, 5, ),

, 2ND,
9

Make sure that U n
is turned off. [Cursor to the = key for
U n
and hit ENTER.]

Press WINDOW to make sure you have settings somewhat close to the following

U n
= V n
= 500 nStart = nMin = 0 nMax = 50
Xmin = 0
Xmax = 50
Xscl = 5
Ymin = 0
Ymax = 600
Yscl = 60

Press GRAPH to display the graph.

Press TRACE and then use the right and left cursor keys to determine the maximum value of the function, V
2
= 570. In this case the contamination increases for n = 0 to n = 2 and then decreases steadily to a limiting value of 0.

Note: If you ever hit the left cursor key during the trace the cursor will jump back to n = 0 and you will have to start the trace again.
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