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TI-83 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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Page 2 of 18
TYPING CONVENTIONS USED IN THIS TUTORIAL
 The keypad portion of a TI-83 is shown at the right.
 The TI-83 sequence keys which we will use extensively in this
tutorial are outlined in white.
 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 would be indicated by “press 2nd, 7”
 Common symbols for this tutorial
 u = 2nd, 7
 v = 2nd, 8
 n = X ,T , ,n
 TBLSET = 2nd, WINDOW
 FORMAT = 2nd, ZOOM
 TABLE = 2nd, GRAPH
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Page 3 of 18
SETTING SEQUENCE MODES FOR THIS TUTORIAL
 The TI-83 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
4th row.
 You may also wish to select Dot (2nd element in the 5th
row) to show discrete values when graphing.
 These setting are shown in the top screen at the right.
Make any other changes as necessary.
 For this tutorial the FORMAT settings should be as shown
in the bottom screen at the right. To do this:
 Press 2nd, ZOOM.
 Use the cursor keys and the ENTER key to make your
settings conform to those shown.
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Page 4 of 18
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 un represent the amount of contaminant in the lake after n-years and the initial amount u0 =
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, u1 = 0.90 u0 + 100.
This suggests that
un = 0.90 un–1 + 100; u0 = 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
ENTERING THE RECURSIVE EQUATION
 Press Y= to enter the sequence equations list.
 Three functions, u(n) , v(n) and w(n) , defined either recursively or explicitly may be entered
along with the appropriate values for nMin, u(nMin), v(nMin) and w(nMin). nMin, u(nMin),
v(nMin) and w(nMin) will be determined by the initial value(s) of the sequence(s). Since u0 =
500 for our sequence: nMin = 0 and u(nMin) = 500.
 Clear any previously entered functions using CLEAR and the cursor keys. [nMin cannot be cleared.
It must be given some value.]
 Also, make sure that the plot style beside u(n) , v(n) and w(n) shows as a  To accomplish this,
cursor to the symbol which isn’t of this form and press ENTER until the symbol  shows.
 Enter u(n) as 0.9 u(n–1) + 100. The keystrokes are:


., 9, 2ND 7, (, n, –, 1, ),+, 1, 0, 0
Also use the cursor keys and the ENTER key to edit
nMin and u(nMin) to the values of 0 and 500
respectively if you have not already done so.
Leave v(n) and w(n) undefined.
 The resulting screen is shown at the right.
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Page 6 of 18
SETTING THE WINDOW and GRAPHING THE RECURSIVE EQUATION
 Press WINDOW to go to the WINDOW setup screen. The complete WINDOW setup screen is
shown in the screen below and at the right (you will have to down cursor to get to the portion below
Xscl).
 General rules for setting WINDOW values
 nMin will default to the value set in the Y= list and vice-versa.
 nMax = a value of your choosing
 PlotStart must always be an integer  1. You must choose 1 if you want the initial value of the
sequence to plot.
 PlotStep determines how far apart the plotted sequence values will be.
 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, the cursor keys 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
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, u0 = 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. Your screen will be similar to the one at
the bottom right. The only difference will be that Un will
be displayed as u(n).
 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
ENTERING THE EXPLICIT EQUATION FOR THE POLLUTION MODEL
 The explicit solution for the recursive equation un = 0.9 un–1 + 100; u0 = 500, n  1 can be shown to be
un = –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 equation 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, ), ^, n, +, 1, 0, 0, 0
 Edit v(nMin) to 500
 The resulting screen is shown at the top right.
 Go to TblSet (2nd, WINDOW) and make sure that TblStart =
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.
(The notations for Un and Vn will be u(n) and v(n).)
 If you were to Press GRAPH at this time you would see just
one graph. Un and Vn may be selectively turned ON or OFF.
This will determine which graph or table is generated.
 Exercise 3: Turn off Un.
Graph Vn for 0  n  50.
Create a table of values for Vn by making TblStart = 0 and Tbl = 1.
Create a second table for Vn with TblStart = 106. (The TI-83 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
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 TI-83 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 difference equations
since ANS will serve as u(n–1).
 Let un = 0.9 un–1 + 100; u0 = 500, n  1
 The following will generate a table of values for u0, u1, u2, …


etc.
(You may need to enter QUIT (2nd, MODE) to get to the Home screen first.) The keystrokes
would be 500, ENTER, ., 9, 2ND (–), +, 1, 0, 0, ENTER, 2nd ENTER, ENTER, 2nd ENTER,
ENTER, etc. [These keystrokes will also give 635.5.]
 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
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 un = 0.9
un–1 + 100; u0 = 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, an, generated by an = 100 (0.75)n – 1; n  1.
The recursive equation for our pollution model is therefore
un = 0.9 un–1 + 100 (0.75)n – 1; u0 = 500, n  1
 Exercise 4: Use a table of values to find u7.
 Click here to Goto Exercise 4 answers.
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Page 11 of 18
THE EXPLICIT SOLUTION
 The explicit equation, obtained again from solution techniques for first order linear difference
equations, is
un = 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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Page 12 of 18
THE FIBONACCI SEQUENCE
 The Fibonacci sequence
u1 = 1, u2 = 1, un = un–1 + un–2 , n  3
is an example of a second order difference equation and can be evaluated using the TI-83.
 Press Y= and make the following definitions:
nMin=1
u(n) = u(n–1)+u(n–2)
u(nMin)={1,1}
The keystrokes are: 2nd (, 1, ,, 1, 2nd )
Clear v(n) and v(nMin)
 Press WINDOW and set the WINDOW as follows:
nMin=1
nMax=10
PlotStart=PlotStep=1
Xmin=1
Xmax=10
Xscl=1
Ymin=0
Ymax=60
Yscl=10
 Press 2nd WINDOW to get to TblSet and make TblStart =
Tbl=1
 Press 2nd GRAPH to get the table shown in the top screen.
u(n) will be the Fibonacci sequence.
 Press GRAPH to see a graph of the sequence.
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Page 13 of 18
Math T E A C H E R Link
TI-83 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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Page 14 of 18
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 a graph similar to the one 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. When you use TRACE the TI-83
scribes the recursive equation u=.9u(n–1)+100 at the top left
of the graphing screen, the values of n and x stacked
vertically at the lower left and Un labeled as Y at the lower
right. (Graphing screens may occasionally be slightly
different as in this case since I have used TI-82 screens where
possible to save composition time.)
Click here to Goto Exercise 1 page.
Page 15 of 18
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 DRAW (2nd PRGM) 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 16 of 18
EXERCISE 3 ANSWERS
 To turn off u(n): go to Y=, cursor to the = for u(n) and press
ENTER so that the = sign is not highlighted (which turns u(n)
off, also u(nMin)). See the screen at the right.
 To graph v(n):
 Go to WINDOW, if necessary, and make sure that 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 2nd
screen at the right.
 To create the first TABLE:
 Go to TblSet (2nd WINDOW)
 Edit TABLE SETUP so that TblStart=0 and Tbl=1 as
shown below.
TABLE SETUP
TblStart=0
Tbl=1
Indpnt:
Auto Ask
Depend:
Auto Ask
 Press TABLE (2nd GRAPH) to get the table shown in the
3rd screen.
 To get the second TABLE:
 Go to TblSet (2nd WINDOW)
 Edit TABLE SETUP so that TblStart=106 and Tbl=1 as
shown below.
TABLE SETUP
TblStart=106
Tbl=1
Indpnt:
Auto Ask
Depend:
Auto Ask
 Press TABLE (2nd GRAPH) to get the table shown in the
4th screen.
Click here to Goto Exercise 3 page
Page 17 of 18
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, (, n, –, 1, ), +, 1, 0, 0,
(, ., 7, 5, ), ^, (n, –, 1, )
 Edit u(nMin) back to a value of 500 if it is not already
there. You may leave v(nMin) at 500 or make it 0.
 To create the TABLE:
 Press 2nd 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 u7 = 469.02.
Click here to Goto Exercise 4 page
Page 18 of 18
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 [You could have also cleared
u(n) and used it instead.]
 The keystrokes are: 1, 1, 6, 6, ., 6, 6, 6, 6, (, ., 9, ),
^, n, –, 6, 6, 6, ., 6, 6, 6, 6, (, ., 7, 5, ), ^, n
 Enter v(nMin)=500
 Make sure that u(n) is turned off. [Cursor to the = key
and hit ENTER.]
 Press WINDOW to make sure you have settings somewhat
close to the following
nMin = 0
nMax = 50
PlotStart=PlotStep=1
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, V2 = 570. In this case the contamination increases for n = 0 to n = 2 and then decreases
steadily to a limiting value of 0.
Click here to Goto Exercise 5 page