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Texas Instruments 83PL/CLM/1L1/V User Manual - Page 186

Recursive Sequences, represented

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For example, in the nonrecursive sequence below, you can calculate u(5) directly, without first calculating u(1) or any previous term. The sequence equation above returns the sequence 2, 4, 6, 8, 10, ...for n = 1, 2, 3, 4, 5, ... . Note: You may leave blank the initial value u(nMin) when calculating nonrecursive sequences. 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). TI-83 Plus Sequence Graphing 183

Section	Page
TI-83 Plus / TI-83 Plus Silver Edition Graphing Calculator Guidebook	1
Important	2
US FCC Information Concerning Radio Frequency Interference	3
Chapter 1: Operating the TI-83 Plus Silver Edition	4
Documentation Conventions	4
TI-83 Plus Keyboard	5
Keyboard Zones	5
TI-83 Plus	6
Using the Color-Coded Keyboard	7
Using the [2nd] and [ALPHA] Keys	7
Turning On and Turning Off the TI-83 Plus	9
Turning On the Calculator	9
Turning Off the Calculator	10
Batteries	10
Setting the Display Contrast	11
Adjusting the Display Contrast	11
When to Replace Batteries	12
The Display	13
Types of Displays	13
Home Screen	13
Displaying Entries and Answers	13
Returning to the Home Screen	14
Busy Indicator	14
Display Cursors	15
Entering Expressions and Instructions	16
What Is an Expression?	16
Entering an Expression	16
Multiple Entries on a Line	17
Entering a Number in Scientific Notation	17
Functions	18
Instructions	19
Interrupting a Calculation	19
TI-83 Plus Edit Keys	20
Setting Modes	22
Checking Mode Settings	22
Changing Mode Settings	23
Setting a Mode from a Program	23
Normal, Sci, Eng	23
Float, 0123456789	24
Radian, Degree	25
Func, Par, Pol, Seq	26
Connected, Dot	26
Sequential, Simul	27
Real, a+bi, re^(theta)i	27
Full, Horiz, G-T	28
Using TI-83 Plus Variable Names	29
Variables and Defined Items	29
Notes about Variables	30
Storing Variable Values	32
Storing Values in a Variable	32
Displaying a Variable Value	33
Archiving Variables (Archive, Unarchive)	33
Recalling Variable Values	34
Using Recall (RCL)	34
ENTRY (Last Entry) Storage Area	36
Using ENTRY (Last Entry)	36
Accessing a Previous Entry	37
Reexecuting the Previous Entry	37
Multiple Entry Values on a Line	38
Clearing ENTRY	39
Using Ans in an Expression	39
Continuing an Expression	40
Storing Answers	41
TI-83 Plus Menus	42
Using a TI-83 Plus Menu	42
Displaying a Menu	43
Moving from One Menu to Another	44
Scrolling a Menu	44
Selecting an Item from a Menu	44
Leaving a Menu without Making a Selection	46
VARS and VARS Y-VARS Menus	47
VARS Menu	47
Selecting a Variable from the VARS Menu or VARS Y-VARS Menu	48
Equation Operating System (EOS)	50
Order of Evaluation	50
Implied Multiplication	51
Parentheses	51
Negation	52
Special Features of the TI-83 Plus	53
Flash – Electronic Upgradability	53
1.56 Megabytes (M) of Available Memory	53
Applications	54
Archiving	54
Calculator-Based Laboratory™ (CBL 2™, CBL™) and	55
Calculator-Based Ranger™ (CBR™)	55
Other TI-83 Plus Features	56
Graphing	56
Sequences	56
Tables	57
Split Screen	57
Matrices	57
Lists	58
Statistics	58
Inferential Statistics	58
Applications	59
CATALOG	59
Programming	60
Archiving	60
Communication Link	60
Error Conditions	62
Diagnosing an Error	62
Correcting an Error	63
Chapter 2: Math, Angle, and Test Operations	64
Getting Started: Coin Flip	64
Keyboard Math Operations	66
Using Lists with Math Operations	66
+ (Addition), - (Subtraction), * (Multiplication), / (Division)	66
Trigonometric Functions	66
^ (Power), ² (Square), (Square Root)	67
(Inverse)	67
log(, 10^(, ln(	68
e^( (Exponential)	68
e (Constant)	68
(Negation)	69
(Pi)	69
MATH Operations	70
MATH Menu	70
> Frac, >Dec	70
³(Cube), (Cube Root)	71
xth (Root)	72
fMin(, fMax(	72
nDeriv(	73
fnInt(	74
Using the Equation Solver	75
Solver	75
Entering an Expression in the Equation Solver	75
Entering and Editing Variable Values	77
Solving for a Variable in the Equation Solver	78
Editing an Equation Stored to eqn	80
Equations with Multiple Roots	80
Further Solutions	80
Controlling the Solution for Solver or solve(	81
Using solve( on the Home Screen or from a Program	81
MATH NUM (Number) Operations	82
MATH NUM Menu	82
abs(	82
round(	83
iPart(, fPart(	83
int(	84
min(, max(	84
lcm(, gcd(	85
Entering and Using Complex Numbers	87
Complex-Number Modes	87
Entering Complex Numbers	88
Note about Radian Versus Degree Mode	88
Interpreting Complex Results	89
Rectangular-Complex Mode	89
Polar-Complex Mode	90
MATH CPX (Complex) Operations	91
MATH CPX Menu	91
conj(	91
real(	92
imag(	92
angle(	93
abs(	93
MATH PRB (Probability) Operations	95
MATH PRB Menu	95
rand	95
nPr, nCr	96
! (Factorial)	97
randInt(	98
randNorm(	98
randBin(	99
ANGLE Operations	100
ANGLE Menu	100
Entry Notation	100
(Degree)	101
(Radians)	102
R>Pr(, R>Pq(, P>Rx(, P>Ry(	103
TEST (Relational) Operations	104
TEST Menu	104
=, not equal, >, gt =, <, lt =	104
Using Tests	105
TEST LOGIC (Boolean) Operations	106
TEST LOGIC Menu	106
Boolean Operators	106
and, or, xor	106
not(	107
Using Boolean Operations	107
Chapter 3: Function Graphing	108
Getting Started: Graphing a Circle	108
Defining Graphs	110
TI-83 Plus—Graphing Mode Similarities	110
Defining a Graph	110
Displaying and Exploring a Graph	111
Saving a Graph for Later Use	111
Setting the Graph Modes	112
Checking and Changing the Graphing Mode	112
Setting Modes from a Program	113
Defining Functions	114
Displaying Functions in the Y= Editor	114
Defining or Editing a Function	114
Defining a Function from the Home Screen or a Program	115
Evaluating Y= Functions in Expressions	116
Selecting and Deselecting Functions	117
Selecting and Deselecting a Function	117
Turning On or Turning Off a Stat Plot in the Y= Editor	118
Selecting and Deselecting Functions from the Home Screen or a Program	118
Setting Graph Styles for Functions	120
Graph Style Icons in the Y= Editor	120
Setting the Graph Style	121
Shading Above and Below	122
Setting a Graph Style from a Program	123
Setting the Viewing Window Variables	124
The TI-83 Plus Viewing Window	124
Displaying the Window Variables	124
Changing a Window Variable Value	125
Storing to a Window Variable from the Home Screen or a Program	126
Delta X and Delta Y	127
Setting the Graph Format	128
Displaying the Format Settings	128
Changing a Format Setting	128
RectGC, PolarGC	129
CoordOn, CoordOff	129
GridOff, GridOn	130
AxesOn, AxesOff	130
LabelOff, LabelOn	130
ExprOn, ExprOff	131
Displaying Graphs	132
Displaying a New Graph	132
Pausing or Stopping a Graph	132
Smart Graph	132
Overlaying Functions on a Graph	133
Graphing a Family of Curves	134
Exploring Graphs with the Free-Moving Cursor	135
Free-Moving Cursor	135
Graphing Accuracy	135
Exploring Graphs with TRACE	137
Beginning a Trace	137
Moving the Trace Cursor	137
Moving the Trace Cursor from Function to Function	138
Moving the Trace Cursor to Any Valid X Value	138
Panning to the Left or Right	139
Quick Zoom	139
Leaving and Returning to TRACE	140
Using TRACE in a Program	140
Exploring Graphs with the ZOOM Instructions	141
ZOOM Menu	141
Zoom Cursor	142
ZBox	142
Zoom In, Zoom Out	143
ZDecimal	144
ZSquare	145
ZStandard	145
ZTrig	145
ZInteger	146
ZoomStat	146
ZoomFit	146
Using ZOOM MEMORY	147
ZOOM MEMORY Menu	147
ZPrevious	147
ZoomSto	147
ZoomRcl	148
ZOOM FACTORS	148
Checking XFact and YFact	148
Changing XFact and YFact	149
Using ZOOM MEMORY Menu Items from the Home Screen or a Program	149
Using the CALC (Calculate) Operations	150
CALCULATE Menu	150
value	150
zero	151
minimum, maximum	153
intersect	154
dy/dx	155
Integral f(x)dx	156
Chapter 4: Parametric Graphing	157
Getting Started: Path of a Ball	157
Defining and Displaying Parametric Graphs	161
TI-83 Plus Graphing Mode Similarities	161
Setting Parametric Graphing Mode	161
Displaying the Parametric Y= Editor	161
Selecting a Graph Style	162
Defining and Editing Parametric Equations	162
Selecting and Deselecting Parametric Equations	163
Setting Window Variables	163
Setting the Graph Format	164
Displaying a Graph	164
Window Variables and Y.VARS Menus	164
Exploring Parametric Graphs	166
Free-Moving Cursor	166
TRACE	166
Moving the Trace Cursor to Any Valid T Value	167
ZOOM	168
CALC	168
Chapter 5: Polar Graphing	169
Getting Started: Polar Rose	169
Defining and Displaying Polar Graphs	171
TI-83 Plus Graphing Mode Similarities	171
Setting Polar Graphing Mode	171
Displaying the Polar Y= Editor	171
Selecting Graph Styles	172
Defining and Editing Polar Equations	172
Selecting and Deselecting Polar Equations	172
Setting Window Variables	173
Setting the Graph Format	174
Displaying a Graph	174
Window Variables and Y-VARS Menus	174
Exploring Polar Graphs	176
Free-Moving Cursor	176
TRACE	176
Moving the Trace Cursor to Any Valid	177
Value	177
ZOOM	177
CALC	178
Chapter 6: Sequence Graphing	179
Getting Started: Forest and Trees	179
Defining and Displaying Sequence Graphs	182
TI-83 Plus Graphing Mode Similarities	182
Setting Sequence Graphing Mode	182
TI-83 Plus Sequence Functions u, v, and w	182
Displaying the Sequence Y= Editor	183
Selecting Graph Styles	184
Selecting and Deselecting Sequence Functions	184
Defining and Editing a Sequence Function	185
Nonrecursive Sequences	185
Recursive Sequences	186
Setting Window Variables	188
Selecting Axes Combinations	190
Setting the Graph Format	190
Setting Axes Format	190
Displaying a Sequence Graph	191
Exploring Sequence Graphs	192
Free-Moving Cursor	192
TRACE	192
Moving the Trace Cursor to Any Valid n Value	193
ZOOM	194
CALC	194
Evaluating u, v, and w	195
Graphing Web Plots	196
Graphing a Web Plot	196
Valid Functions for Web Plots	196
Displaying the Graph Screen	197
Drawing the Web	197
Using Web Plots to Illustrate Convergence	198
Example: Convergence	198
Graphing Phase Plots	200
Graphing with uv, vw, and uw	200
Example: Predator-Prey Model	200
Comparing TI-83 Plus and TI-82 Sequence Variables	204
Sequences and Window Variables	204
Keystroke Differences Between TI-83 Plus and TI- 82	205
Sequence Keystroke Changes	205
Chapter 7: Tables	206
Getting Started: Roots of a Function	206
Setting Up the Table	208
TABLE SETUP Screen	208
TblStart, Delta Tbl	208
Indpnt: Auto, Indpnt: Ask, Depend: Auto, Depend: Ask	209
Setting Up the Table from the Home Screen or a Program	209
Defining the Dependent Variables	210
Defining Dependent Variables from the Y= Editor	210
Editing Dependent Variables from the Table Editor	210
Displaying the Table	212
The Table	212
Independent and Dependent Variables	212
Clearing the Table from the Home Screen or a Program	213
Scrolling Independent-Variable Values	213
Displaying Other Dependent Variables	214
Chapter 8: Draw Instructions	215
Getting Started: Drawing a Tangent Line	215
Using the DRAW Menu	217
DRAW Menu	217
Before Drawing on a Graph	218
Drawing on a Graph	218
Clearing Drawings	220
Clearing Drawings When a Graph Is Displayed	220
Clearing Drawings from the Home Screen or a Program	220
Drawing Line Segments	221
Drawing a Line Segment Directly on a Graph	221
Drawing a Line Segment from the Home Screen or a Program	222
Drawing Horizontal and Vertical Lines	223
Drawing a Line Directly on a Graph	223
Drawing a Line from the Home Screen or a Program	224
Drawing Tangent Lines	225
Drawing a Tangent Line Directly on a Graph	225
Drawing a Tangent Line from the Home Screen or a Program	226
Drawing Functions and Inverses	227
Drawing a Function	227
Drawing an Inverse of a Function	227
Shading Areas on a Graph	229
Shading a Graph	229
Drawing Circles	231
Drawing a Circle Directly on a Graph	231
Drawing a Circle from the Home Screen or a Program	232
Placing Text on a Graph	233
Placing Text Directly on a Graph	233
Placing Text on a Graph from the Home Screen or a Program	233
Split Screen	234
Using Pen to Draw on a Graph	235
Using Pen to Draw on a Graph	235
Drawing Points on a Graph	236
DRAW POINTS Menu	236
Drawing Points Directly on a Graph with Pt-On(	236
Erasing Points with Pt-Off(	237
Changing Points with Pt-Change(	238
Drawing Points from the Home Screen or a Program	238
Drawing Pixels	240
TI-83 Plus Pixels	240
Turning On and Off Pixels with Pxl-On( and Pxl-Off(	240
Using pxl-Test(	241
Split Screen	241
Storing Graph Pictures (Pic)	242
DRAW STO Menu	242
Storing a Graph Picture	242
Recalling Graph Pictures (Pic)	244
Recalling a Graph Picture	244
Deleting a Graph Picture	244
Storing Graph Databases (GDB)	245
What Is a Graph Database?	245
Storing a Graph Database	245
Recalling Graph Databases (GDB)	247
Recalling a Graph Database	247
Deleting a Graph Database	248
Chapter 9: Split Screen	249
Getting Started: Exploring the Unit Circle	249
Using Split Screen	251
Setting a Split-Screen Mode	251
Horiz (Horizontal) Split Screen	253
Horiz Mode	253
Moving from Half to Half in Horiz Mode	254
Full Screens in Horiz Mode	254
G.T (Graph-Table) Split Screen	255
G-T Mode	255
Moving from Half to Half in G-T Mode	255
Using [TRACE] in G-T Mode	256
Full Screens in G-T Mode	256
TI-83 Plus Pixels in Horiz and G-T Modes	257
TI-83 Plus Pixels in Horiz and G-T Modes	257
DRAW POINTS Menu Pixel Instructions	257
DRAW Menu Text( Instruction	257
PRGM I/O Menu Output( Instruction	258
Setting a Split-Screen Mode from the Home Screen or a Program	258
Chapter 10: Matrices	259
Getting Started: Systems of Linear Equations	259
Defining a Matrix	261
What Is a Matrix?	261
Selecting a Matrix	261
Accepting or Changing Matrix Dimensions	262
Viewing and Editing Matrix Elements	263
Displaying Matrix Elements	263
Deleting a Matrix	264
Viewing a Matrix	264
Viewing-Context Keys	265
Editing a Matrix Element	265
Editing-Context Keys	267
Using Matrices with Expressions	268
Using a Matrix in an Expression	268
Entering a Matrix in an Expression	268
Displaying and Copying Matrices	270
Displaying a Matrix	270
Copying One Matrix to Another	270
Accessing a Matrix Element	271
Using Math Functions with Matrices	272
Using Math Functions with Matrices	272
+ (Add), – (Subtract), * (Multiply)	272
(Negation)	273
abs(	274
round(	274
(Inverse)	274
Powers	275
Relational Operations	275
iPart(, fPart(, int(	276
Using the MATRX MATH Operations	277
MATRX MATH Menu	277
det(	278
(Transpose)	278
Accessing Matrix Dimensions with dim(	278
Creating a Matrix with dim(	279
Redimensioning a Matrix with dim(	279
Fill(	279
identity(	280
randM(	280
augment(	280
Matr>list(	281
List>matr(	282
cumSum(	282
Row Operations	283
ref(, rref(	283
rowSwap(	284
row+(	284
Chapter 11: Lists	286
Getting Started: Generating a Sequence	286
Naming Lists	289
Using TI-83 Plus List Names L1 through L6	289
Creating a List Name on the Home Screen	289
Storing and Displaying Lists	291
Storing Elements to a List	291
Displaying a List on the Home Screen	291
Copying One List to Another	292
Accessing a List Element	292
Deleting a List from Memory	292
Using Lists in Graphing	293
Entering List Names	294
Using the LIST NAMES Menu	294
Entering a User-Created List Name Directly	295
Attaching Formulas to List Names	296
Attaching a Formula to a List Name	296
Attaching a Formula to a List on the Home Screen or in a Program	297
Detaching a Formula from a List	299
Using Lists in Expressions	300