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TI82 SEQUENCE TUTORIAL—AN OVERVIEW

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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TYPING CONVENTIONS USED IN THIS TUTORIAL

The TI82 sequence keys which we will use extensively in this tutorial are outlined in blue on the screen shown at the right.

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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SETTING SEQUENCE MODES FOR THIS TUTORIAL

The TI82 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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EXAMPLE 1—A POLLUTION MODEL

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 nyears and the initial amount u_{0} = 500.

After 1year 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 nyears.
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ENTERING THE RECURSIVE EQUATION

Press Y= to enter the sequence equations list.

Two functions, U_{n} and V_{n}, defined either recursively or explicitly may be entered.

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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SETTING THE WINDOW and GRAPHING THE RECURSIVE EQUATION

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 UnStart, VnStart and nStart, u_{0} = 500 for our example.

nMin = nStart unless you want the graph to start at a sequence value different from the sequence value at nStart.

nMax = a value of your choosing

Xmin, Xmax = nMin, nMax 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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GENERATING A TABLE OF VALUES FOR THE RECURSIVE EQUATION

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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ENTERING THE EXPLICIT EQUATION FOR THE POLLUTION MODEL

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
(–), 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 TI82 will grind for quite a while before you will see the table.)

Click here to Goto Exercise 3 answers.
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A SPECIAL NOTE FOR TEACHERS
CREATING TABLE VALUES USING THE ‘ANS’ FUNCTION

To help students understand how values for difference equations are generated I have them calculate sequence values by hand using the TI82 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}, …

TI82 home screen

Keystrokes

500

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}]


In fact this is the approach I use entirely in my classes as some students are still using the TI81 which does not have sequence capabilities.
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ANOTHER POLLUTION MODEL

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_{7}.

Click here to Goto Exercise 4 answers.
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THE EXPLICIT SOLUTION

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.

Click here to Goto Exercise 5 answers
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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 TI82 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

1

1

2

1

2

3

2

3

4

3

5

5

5

8

6

8

13

7

13

21

Press Y=, turnoff 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 TI82 screen.]
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Math Link
TI82 Sequence Tutorial
The End
We hope you enjoyed it!
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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EXERCISE 1 ANSWERS

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.
EXERCISE 2 ANSWERS

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.
EXERCISE 3 ANSWERS

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.

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
EXERCISE 4 ANSWERS

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.
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EXERCISE 5 ANSWERS

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)^{n}

The keystrokes are: 1, 1, 6, 6, ., 6, 6, 6, 6, (, ., 9, ), , 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.
Click here to Goto Exercise 5 page 