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Texas Instruments TI 84 User Guide - Page 161

Recursive Sequences, the sequence above returns 1

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Recursive Sequences In a recursive sequence, the nth term in the sequence is defined in relation to the previous term or the term that precedes the previous term, represented by u(nN1) and u(nN2). A recursive sequence may also be defined in relation to n, as in u(n)=u(nN1)+n. For example, in the sequence below you cannot calculate u(5) without first calculating u(1), u(2), u(3), and u(4). Using an initial value u(nMin) = 1, the sequence above returns 1, 2, 4, 8, 16, ... . Note: On the TI-84 Plus, you must type each character of the terms. For example, to enter u(nN1), press y [u Recursive sequences require an initial value or values, since they reference undefined terms. • If each term in the sequence is defined in relation to the previous term, as in u(nN1), you must specify an initial value for the first term. • If each term in the sequence is defined in relation to the term that precedes the previous term, as in u(nN2), you must specify initial values for the first two terms. Chapter 6: Sequence Graphing 158

Section	Page
TI-84 Plus TI-84 Plus Silver Edition Guidebook	1
Important Information	2
USA FCC Information Concerning Radio Frequency Interference	3
Chapter 1: Operating the TI-84 Plus Silver Edition	4
Documentation Conventions	4
TI-84 Plus Keyboard	4
Keyboard Zones	4
TI-84 Plus	5
Using the Color.Coded Keyboard	6
Using the y and É Keys	6
Turning On and Turning Off the TI-84 Plus	7
Turning On the Graphing Calculator	7
Turning Off the Graphing Calculator	8
Batteries	8
Setting the Display Contrast	9
Adjusting the Display Contrast	9
When to Replace Batteries	10
The Display	10
Types of Displays	10
Home Screen	10
Displaying Entries and Answers	11
Returning to the Home Screen	11
Busy Indicator	12
Display Cursors	12
Interchangeable Faceplates	13
Removing a Faceplate	13
Installing New Faceplates	14
Using the Clock	14
Displaying the Clock Settings	15
Changing the Clock settings	15
Error Messages	16
Turning the Clock On	16
Turning the Clock Off	17
Entering Expressions and Instructions	18
What Is an Expression?	18
Entering an Expression	18
Multiple Entries on a Line	19
Entering a Number in Scientific Notation	19
Functions	20
Instructions	20
Interrupting a Calculation	20
TI-84 Plus Edit Keys	21
Setting Modes	22
Checking Mode Settings	22
Changing Mode Settings	23
Setting a Mode from a Program	24
Normal, Sci, Eng	24
Float, 0123456789	25
Radian, Degree	25
Func, Par, Pol, Seq	26
Connected, Dot	26
Sequential, Simul	26
Real, a+bi, re^qi	27
Full, Horiz, G-T	27
Set Clock	27
Using TI-84 Plus Variable Names	28
Variables and Defined Items	28
Notes about Variables	29
Storing Variable Values	30
Storing Values in a Variable	30
Displaying a Variable Value	30
Archiving Variables (Archive, Unarchive)	31
Recalling Variable Values	31
Using Recall (RCL)	31
ENTRY (Last Entry) Storage Area	32
Using ENTRY (Last Entry)	32
Accessing a Previous Entry	33
Reexecuting the Previous Entry	34
Multiple Entry Values on a Line	34
Clearing ENTRY	35
Using Ans in an Expression	35
Continuing an Expression	36
Storing Answers	36
TI-84 Plus Menus	37
Using a TI-84 Plus Menu	37
Displaying a Menu	38
Moving from One Menu to Another	39
Scrolling a Menu	39
Selecting an Item from a Menu	40
Leaving a Menu without Making a Selection	41
VARS and VARS Y-VARS Menus	41
VARS Menu	41
Selecting a Variable from the VARS Menu or VARS Y-VARS Menu	42
Equation Operating System (EOS™)	43
Order of Evaluation	43
Implied Multiplication	44
Parentheses	44
Negation	45
Special Features of the TI-84 Plus	46
Flash - Electronic Upgradability	46
1.5 Megabytes (M) of Available Memory	46
Applications	47
Archiving	47
Other TI-84 Plus Features	47
Graphing	48
Sequences	48
Tables	48
Split Screen	48
Matrices	49
Lists	49
Statistics	49
Inferential Statistics	50
Applications	50
CATALOG	50
Programming	50
Archiving	51
Communication Link	51
Error Conditions	52
Diagnosing an Error	52
Correcting an Error	53
Chapter 2: Math, Angle, and Test Operations	54
Getting Started: Coin Flip	54
Keyboard Math Operations	55
Using Lists with Math Operations	55
Addition, Subtraction, Multiplication, Division	56
Trigonometric Functions	56
Power, Square, Square Root	57
Inverse	57
log(, 10^(, ln(	57
Exponential	58
Constant	58
Negation	58
Pi	59
MATH Operations	60
MATH Menu	60
4Frac, 4Dec	60
Cube, Cube Root	61
xá (Root)	62
fMin(, fMax(	62
nDeriv(	63
fnInt(	64
Using the Equation Solver	64
Solver	64
Entering an Expression in the Equation Solver	65
Entering and Editing Variable Values	66
Solving for a Variable in the Equation Solver	67
Editing an Equation Stored to eqn	69
Equations with Multiple Roots	69
Further Solutions	69
Controlling the Solution for Solver or solve(	69
Using solve( on the Home Screen or from a Program	70
MATH NUM (Number) Operations	71
MATH NUM Menu	71
abs(	71
round(	72
iPart(, fPart(	72
int(	73
min(, max(	73
lcm(, gcd(	74
Entering and Using Complex Numbers	75
Complex-Number Modes	75
Entering Complex Numbers	76
Note about Radian Versus Degree Mode	76
Interpreting Complex Results	77
Rectangular-Complex Mode	77
Polar-Complex Mode	78
MATH CPX (Complex) Operations	79
MATH CPX Menu	79
conj(	79
real(	80
imag(	80
angle(	80
abs(	81
4Rect	81
4Polar	82
MATH PRB (Probability) Operations	82
MATH PRB Menu	82
rand	83
nPr, nCr	83
Factorial	84
randInt(	85
randNorm(	85
randBin(	86
ANGLE Operations	87
ANGLE Menu	87
Entry Notation	87
Degree	88
Radians	88
8DMS	89
R8Pr(, R8Pq(, P8Rx(, P8Ry(	89
TEST (Relational) Operations	91
TEST Menu	91
ƒ=, É, >, Ç, <, Å	91
Using Tests	92
TEST LOGIC (Boolean) Operations	93
TEST LOGIC Menu	93
Boolean Operators	93
and, or, xor	93
not(	94
Using Boolean Operations	94
Chapter 3: Function Graphing	95
Getting Started: Graphing a Circle	95
Defining Graphs	96
TI-84 Plus-Graphing Mode Similarities	96
Defining a Graph	97
Displaying and Exploring a Graph	97
Saving a Graph for Later Use	97
Setting the Graph Modes	98
Checking and Changing the Graphing Mode	98
Setting Modes from a Program	99
Defining Functions	99
Displaying Functions in the Y= Editor	99
Defining or Editing a Function	100
Defining a Function from the Home Screen or a Program	101
Evaluating Y= Functions in Expressions	101
Selecting and Deselecting Functions	102
Selecting and Deselecting a Function	102
Turning On or Turning Off a Stat Plot in the Y= Editor	103
Selecting and Deselecting Functions from the Home Screen or a Program	103
Setting Graph Styles for Functions	104
MATH Graph Style Icons in the Y= Editor	104
Setting the Graph Style	105
Shading Above and Below	106
Setting a Graph Style from a Program	107
Setting the Viewing Window Variables	108
The TI-84 Plus Viewing Window	108
Displaying the Window Variables	108
Changing a Window Variable Value	109
Storing to a Window Variable from the Home Screen or a Program	109
@X and @Y	110
Setting the Graph Format	111
Displaying the Format Settings	111
Changing a Format Setting	111
RectGC, PolarGC	112
CoordOn, CoordOff	112
GridOff, GridOn	112
AxesOn, AxesOff	113
LabelOff, LabelOn	113
ExprOn, ExprOff	113
Displaying Graphs	114
Displaying a New Graph	114
Pausing or Stopping a Graph	114
Smart Graph	114
Overlaying Functions on a Graph	115
Graphing a Family of Curves	115
Exploring Graphs with the Free-Moving Cursor	116
Free-Moving Cursor	116
Graphing Accuracy	117
Exploring Graphs with TRACE	118
Beginning a Trace	118
Moving the Trace Cursor	118
Moving the Trace Cursor from Function to Function	119
Moving the Trace Cursor to Any Valid X Value	119
Panning to the Left or Right	120
Quick Zoom	120
Leaving and Returning to TRACE	120
Using TRACE in a Program	121
Exploring Graphs with the ZOOM Instructions	121
ZOOM Menu	121
Zoom Cursor	122
ZBox	122
Zoom In, Zoom Out	123
ZDecimal	124
ZSquare	124
ZStandard	125
ZTrig	125
ZInteger	125
ZoomStat	126
ZoomFit	126
Using ZOOM MEMORY	126
ZOOM MEMORY Menu	126
ZPrevious	127
ZoomSto	127
ZoomRcl	127
ZOOM FACTORS	127
Checking XFact and YFact	128
Changing XFact and YFact	128
Using ZOOM MEMORY Menu Items from the Home Screen or a Program	128
Using the CALC (Calculate) Operations	129
CALCULATE Menu	129
value	129
zero	130
minimum, maximum	132
intersect	132
dy/dx	133
âf(x)dx	134
Chapter 4: Parametric Graphing	136
Getting Started: Path of a Ball	136
Defining and Displaying Parametric Graphs	139
TI-84 Plus Graphing Mode Similarities	139
Setting Parametric Graphing Mode	140
Displaying the Parametric Y= Editor	140
Selecting a Graph Style	140
Defining and Editing Parametric Equations	141
Selecting and Deselecting Parametric Equations	141
Setting Window Variables	142
Setting the Graph Format	142
Displaying a Graph	143
Window Variables and Y.VARS Menus	143
Exploring Parametric Graphs	144
Free-Moving Cursor	144
TRACE	144
Moving the Trace Cursor to Any Valid T Value	145
ZOOM	146
CALC	146
Chapter 5: Polar Graphing	147
Getting Started: Polar Rose	147
Defining and Displaying Polar Graphs	148
TI-84 Plus Graphing Mode Similarities	148
Setting Polar Graphing Mode	149
Displaying the Polar Y= Editor	149
Selecting Graph Styles	149
Defining and Editing Polar Equations	150
Selecting and Deselecting Polar Equations	150
Setting Window Variables	150
Setting the Graph Format	151
Displaying a Graph	151
Window Variables and Y.VARS Menus	152
Exploring Polar Graphs	153
Free-Moving Cursor	153
TRACE	153
Moving the Trace Cursor to Any Valid Theta Value	154
ZOOM	154
CALC	154
Chapter 6: Sequence Graphing	155
Getting Started: Forest and Trees	155
Defining and Displaying Sequence Graphs	157
TI-84 Plus Graphing Mode Similarities	157
Setting Sequence Graphing Mode	157
TI-84 Plus Sequence Functions u, v, and w	157
Displaying the Sequence Y= Editor	158
Selecting Graph Styles	159
Selecting and Deselecting Sequence Functions	159
Defining and Editing a Sequence Function	159
Nonrecursive Sequences	160
Recursive Sequences	161
Setting Window Variables	162
Selecting Axes Combinations	164
Setting the Graph Format	164
Setting Axes Format	164
Displaying a Sequence Graph	165
Exploring Sequence Graphs	165
Free-Moving Cursor	165
TRACE	165
Moving the Trace Cursor to Any Valid n Value	166
ZOOM	167
CALC	167
Evaluating u, v, and w	168
Graphing Web Plots	168
Graphing a Web Plot	168
Valid Functions for Web Plots	169
Displaying the Graph Screen	169
Drawing the Web	169
Using Web Plots to Illustrate Convergence	170
Example: Convergence	170
Graphing Phase Plots	172
Graphing with uv, vw, and uw	172
Example: Predator-Prey Model	172
Comparing TI-84 Plus and TI-82 Sequence Variables	175
Sequences and Window Variables	175
Keystroke Differences Between TI-84 Plus and TI-82	176
Sequence Keystroke Changes	176
Chapter 7: Tables	178
Getting Started: Roots of a Function	178
Setting Up the Table	179
TABLE SETUP Screen	179
TblStart, @Tbl	180
Indpnt: Auto, Indpnt: Ask, Depend: Auto, Depend: Ask	180
Setting Up the Table from the Home Screen or a Program	181
Defining the Dependent Variables	181
Defining Dependent Variables from the Y= Editor	181
Editing Dependent Variables from the Table Editor	181
Displaying the Table	183
The Table	183
Independent and Dependent Variables	183
Clearing the Table from the Home Screen or a Program	184
Scrolling Independent-Variable Values	184
Displaying Other Dependent Variables	185
Chapter 8: Draw Instructions	187
Getting Started: Drawing a Tangent Line	187
Using the DRAW Menu	189
DRAW Menu	189
Before Drawing on a Graph	190
Drawing on a Graph	190
Clearing Drawings	191
Clearing Drawings When a Graph Is Displayed	191
Clearing Drawings from the Home Screen or a Program	191
Drawing Line Segments	192
Drawing a Line Segment Directly on a Graph	192
Drawing a Line Segment from the Home Screen or a Program	192
Drawing Horizontal and Vertical Lines	193
Drawing a Line Directly on a Graph	193
Drawing a Line from the Home Screen or a Program	194
Drawing Tangent Lines	195
Drawing a Tangent Line Directly on a Graph	195
Drawing a Tangent Line from the Home Screen or a Program	196
Drawing Functions and Inverses	197
Drawing a Function	197
Drawing an Inverse of a Function	197
Shading Areas on a Graph	198
Shading a Graph	198
Drawing Circles	200
Drawing a Circle Directly on a Graph	200
Drawing a Circle from the Home Screen or a Program	200
Placing Text on a Graph	201
Placing Text Directly on a Graph	201
Placing Text on a Graph from the Home Screen or a Program	202
Split Screen	202
Using Pen to Draw on a Graph	203
Using Pen to Draw on a Graph	203
Drawing Points on a Graph	204
DRAW POINTS Menu	204
Drawing Points Directly on a Graph with Pt-On(	204
Erasing Points with Pt-Off(	205
Changing Points with Pt-Change(	205
Drawing Points from the Home Screen or a Program	206
Drawing Pixels	206
TI-84 Plus Pixels	206
Turning On and Off Pixels with Pxl-On( and Pxl-Off(	207
Using pxl-Test(	207
Split Screen	208
Storing Graph Pictures (Pic)	208
DRAW STO Menu	208
Storing a Graph Picture	209
Recalling Graph Pictures (Pic)	209
Recalling a Graph Picture	209
Deleting a Graph Picture	210
Storing Graph Databases (GDB)	210
What Is a Graph Database?	210
Storing a Graph Database	211
Recalling Graph Databases (GDB)	212
Recalling a Graph Database	212
Deleting a Graph Database	212
Chapter 9: Split Screen	214
Getting Started: Exploring the Unit Circle	214
Using Split Screen	216
Setting a Split-Screen Mode	216
Horiz (Horizontal) Split Screen	218
Horiz Mode	218
Moving from Half to Half in Horiz Mode	218
Full Screens in Horiz Mode	219
G-T (Graph-Table) Split Screen	219
G-T Mode	219
Moving from Half to Half in G-T Mode	220
Using TRACE in G-T Mode	220
Full Screens in G-T Mode	221
TI-84 Plus Pixels in Horiz and G-T Modes	221
TI-84 Plus Pixels in Horiz and G-T Modes	221
DRAW POINTS Menu Pixel Instructions	222
DRAW Menu Text( Instruction	222
PRGM I/O Menu Output( Instruction	222
Setting a Split-Screen Mode from the Home Screen or a Program	223
Chapter 10: Matrices	224
Getting Started: Systems of Linear Equations	224
Defining a Matrix	225
What Is a Matrix?	225
Selecting a Matrix	226
Accepting or Changing Matrix Dimensions	226
Viewing and Editing Matrix Elements	227
Displaying Matrix Elements	227
Deleting a Matrix	228
Viewing a Matrix	228
Viewing-Context Keys	228
Editing a Matrix Element	229
Editing-Context Keys	230
Using Matrices with Expressions	231
Using a Matrix in an Expression	231
Entering a Matrix in an Expression	231
Displaying and Copying Matrices	232
Displaying a Matrix	232
Copying One Matrix to Another	233
Accessing a Matrix Element	233
Using Math Functions with Matrices	234
Using Math Functions with Matrices	234
Addition, Subtraction, Multiplication	234
Negation	235
abs(	236
round(	236
Inverse	236
Powers	237
Relational Operations	237
iPart(, fPart(, int(	238
Using the MATRX MATH Operations	239
MATRX MATH Menu	239
det(	240
Transpose	240
Accessing Matrix Dimensions with dim(	241
Creating a Matrix with dim(	241
Redimensioning a Matrix with dim(	242
Fill(	242
identity(	242
randM(	243
augment(	243
Matr4list(	243
List4matr(	244
cumSum(	245
Row Operations	245
ref(, rref(	246
rowSwap(	246
row+(	247
Örow(	247
Örow+(	247
Chapter 11: Lists	249
Getting Started: Generating a Sequence	249
Naming Lists	250
Using TI-84 Plus List Names L1 through L6	250
Creating a List Name on the Home Screen	251
Storing and Displaying Lists	252
Storing Elements to a List	252
Displaying a List on the Home Screen	253