INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Geometric Sequences

Slide Duration:

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
0:18
Same Dimensions
0:25
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities

31m 48s

Intro
0:00
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
0:16
Standard Form
0:18
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
0:21
Standard Form
0:29
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
0:11
Test Point
0:18
0:29
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
8:22
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
3:24
Radicand No nth Powers
3:47
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
16:12
Same Index, Same Radicand
16:20
16:28
19:04
Distributive Property
19:10
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
0:11
0:22
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
11:27
Restriction: Index is Even
11:53
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
16:54
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50

47m 4s

Intro
0:00
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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• ## Related Books 1 answer Last reply by: Dr Carleen EatonSun Oct 21, 2012 7:55 PMPost by Joe Snyder on October 15, 2012I need help with finding the sum of a geo sequence

### Geometric Sequences

• You can use the formula for the nth term of a series to find the first term.
• If you need to find a particular term and you are given the kth term, first use the kth term to find the first term. Then use the first term to find the term you are asked to find.
• When finding geometric means, remember that you might get two different values of r, one positive and the other negative. Use both values to obtain two different sets of geometric means. Each set is a solution to the problem.

### Geometric Sequences

Find the 10th term of the geometric sequence with a6 = − 3072 , r = 4
• Step 1 - Find a1 using the formula an = a1*(r)n − 1 and the information given
• an = a1*(r)n − 1
• a6 = a1*(4)6 − 1
• − 3072 = a1*(4)5
• − 3072 = a1*1024
• a1 = − 3
• Step 2 - Using a1 find the explicit formula, then find the 10th term.
• an = a1*(r)n − 1
• an = ( − 3)*(4)n − 1
• a10 = ( − 3)*(4)10 − 1
• a10 = ( − 3)*(4)9
• a10 = ( − 3)*(262144)
a10 = − 786432
Find the 9th term of the geometric sequence with a2 = − 8 , r = 2
• Step 1 - Find a1 using the formula an = a1*(r)n − 1 and the information given
• an = a1*(r)n − 1
• a2 = a1*(2)2 − 1
• − 8 = a1*(2)1
• − 8 = a1*2
• a1 = − 4
• Step 2 - Using a1 find the explicit formula, then find the 10th term.
• an = a1*(r)n − 1
• an = ( − 4)*(2)n − 1
• a9 = ( − 4)*(2)9 − 1
• a9 = ( − 4)*(2)8
• a9 = ( − 4)*(256)
a9 = − 1024
Find the 12th term of the geometric sequence with a6 = 243 , r = 3
• Step 1 - Find a1 using the formula an = a1*(r)n − 1 and the information given
• an = a1*(r)n − 1
• a6 = a1*(3)6 − 1
• 243 = a1*(3)5
• 243 = a1*243
• a1 = 1
• Step 2 - Using a1 find the explicit formula, then find the 10th term.
• an = a1*(r)n − 1
• an = (1)*(3)n − 1
• a12 = (1)*(3)12 − 1
• a12 = (1)*(3)11
• a12 = (1)*(177147)
a12 = 177147
Write an equation for the nth term of the geometric sequence 4,12,36,108,...
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [12/4] = 3
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
an = 4*(3)n − 1
Write an equation for the nth term of the geometric sequence 4,8,16,32,...
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [8/4] = 2
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
an = 4*(2)n − 1
Write an equation for the nth term of the geometric sequence − 4,12, − 36,108,...
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [12/( − 4)] = − 3
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
an = − 4*( − 3)n − 1
Write an equation for the nth term of the geometric sequence − 4, − 8, − 16, − 32,...
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [( − 8)/( − 4)] = 2
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
an = − 4*(2)n − 1
Write the next 3 terms of the geometric sequence 3,6,12,24
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [6/3] = 2
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
• an = 3*(2)n − 1
• Step 3) Find the next three terms by evaluating a5,a6,a7
• a5 =
• a6 =
• a7 =
• a5 = 3*(2)5 − 1 = 3*(2)4 = 3*16 = 48
• a6 = 3*(2)6 − 1 = 3*(2)5 = 3*32 = 96
a7 = 3*(2)7 − 1 = 3*(2)6 = 3*64 = 192
Write the next 3 terms of the geometric sequence 3,18,108,648
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [18/3] = 6
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
• an = 3*(6)n − 1
• Step 3) Find the next three terms by evaluating a5,a6,a7
• a5 =
• a6 =
• a7 =
• a5 = 3*(6)5 − 1 = 3*(6)4 = 3*1296 = 3888
• a6 = 3*(6)6 − 1 = 3*(6)5 = 3*7776 = 23328
a7 = 3*(6)7 − 1 = 3*(6)6 = 3*46656 = 139968
Write the next 3 terms of the geometric sequence 4, − 8,16, − 32
• Step 1) Find the common ration by dividing the second term by the first term
• r = [(a2)/(a1)] = [( − 8)/4] = − 2
• Step 2 )Write the explicit/general formula using the formula an = a1*(r)n − 1
• an = a1*(r)n − 1
• an = 4*( − 2)n − 1
• Step 3) Find the next three terms by evaluating a5,a6,a7
• a5 =
• a6 =
• a7 =
• a5 = 4*(6)5 − 1 = 4*( − 2)4 = 4*16 = 64
• a6 = 4*(6)6 − 1 = 4*( − 2)5 = 4*( − 32) = − 128
a7 = 4*(6)7 − 1 = 4*( − 2)6 = 4*64 = 256

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Geometric Sequences

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Geometric Sequences 0:11
• Common Difference
• Common Ratio
• Example: Geometric Sequence
• nth Term of a Geometric Sequence 4:41
• Example: nth Term
• Geometric Means 6:51
• Example: Geometric Mean
• Example 1: 9th Term 12:04
• Example 2: Geometric Means 15:18
• Example 3: nth Term 18:32
• Example 4: Three Terms 20:59

### Transcription: Geometric Sequences

Welcome to Educator.com.0000

In the previous lessons, we talked about arithmetic sequences and series.0002

So, we are going to go on to discuss geometric sequences.0007

What are geometric sequences? Recall that a sequence in general is a list of numbers in a certain order.0012

So, it is in the general form first term, second term, and on...and it could end at a particular term, an, or it may go on indefinitely.0020

In previous lessons, we talked about arithmetic sequences: for example, 5, 10, 15, 20.0034

And each term was related to the previous one by a common difference, d (here d = 5).0043

What we did is added whatever the common difference was to a term to get the next, to get the next, and so on.0051

So, if you need to review arithmetic sequences or sequences in general, it might be a good idea to go back and start with that lecture.0057

And now, we are going to continue on and learn about geometric sequences.0064

So again, a geometric sequence is a list of numbers; so a geometric sequence is a sequence0067

in which each term after the first is found by multiply the previous term by a non-zero constant r.0073

When we talked about arithmetic sequences, we added the common difference.0081

Here we are going to multiply a term by the common ratio r to get the next term.0087

Therefore, an equals the previous term, which is an - 1, times the common ratio.0092

So, if I were to look for a4 in a particular series, I would find it by saying,0103

"OK, the previous term, a4 - 1, times the common ratio..." so the term a4 would equal r times the third term.0109

Looking at an example of this, this, again, was an arithmetic sequence.0124

Now, we are going on and talking about an example that gives you a geometric sequence: 3, 12, 48, 192.0150

Working with these, it is important to find a common ratio: the common ratio is given by taking a term0166

(any term--I am going to take 12) and dividing it by the previous term.0173

So, this is a2; the first term is a1; therefore, if I take a2/a2 - 1, that is just a2/a1.0178

So, take a term; divide it by the previous term; and this is going to give me the common ratio of 4.0191

So, you can see where these two equations come from.0206

To find the next term, you multiply the term previous to it by the common ratio.0209

To find the common ratio, you take a term, and you divide it by the term that came just before.0215

Now, as we talked about arithmetic sequences, I said that the common difference could be a negative number; it could also be a fraction.0223

And that is true, as well, with the geometric series.0231

For example, I could have 2, -6, 18, -54: this is another geometric sequence, and I want to find the common ratio.0237

So, I will take a term; I am going to go for -6, and I am going to divide it by the previous term, which is 2; and it is going to give me -3.0250

The thing that you will notice is: when you have a negative common ratio, the terms are going to alternate their signs.0257

So, I am going to have a positive, a negative, a positive, a negative.0265

If it is a positive common ratio, then the terms will just all be positive.0269

You can also have a common ratio that is a fraction; and we will work with those examples later on in the lesson.0274

For a geometric sequence, there is a formula for the nth term.0282

So here, the formula for the nth term is that the general term, an,0286

equals the first term, times the common ratio, taken to the power n - 1.0290

Looking at a geometric sequence: 2, 6, 18, and we will make this an infinite sequence...0296

it is going to go on and on...perhaps I was asked to find the sixth term, a6.0302

What is that? Well, I can use this formula.0309

a6 equals the first term (I have the first term--it is 2), and I also need to find the common ratio.0312

The common ratio is going to be any of the terms (I will take 18), divided by the previous term (6), which is 3.0320

So, I have the common ratio; I have the first term; and I have n; here, since I am looking for the sixth term, n will equal 6.0329

So, the sixth term is equal to the first term (which is 2), time the common ratio (which is 3), raised to the power of 6 - 1.0339

The sixth term is equal to 2, times 35.0354

So, recalling powers of 3: 3 times 3 is 9, times 3 is 27; so 33 is 27; 34 is 81; and 35 is 243, 81 times 3.0359

So, a6 = 2 times 243, or simply 486.0381

So again, this is the formula for the nth term of a geometric sequence.0389

Given the sequence, I could find the sixth term, because I know the first term, 2; I know the common ratio;0394

I was able to figure out that it is 3; and I know that n is equal to 6; so I got that a6 is 486.0404

Geometric means: again, thinking back to arithmetic sequences, we said that arithmetic means are missing terms in an arithmetic series.0412

And that is analogous to this situation: geometric means are missing terms between two non-successive terms of a geometric sequence.0421

2048 is the first term; then I have three missing terms--those are the geometric means; and then, I have my last term, 8.0429

Use the common ratio to find the geometric means.0442

If I find the common ratio, then all I need to do is take a term and multiply it by the common ratio to find the next term;0444

multiply that by the common ratio to find the next term; and so on.0450

If I want to find the missing terms here, I am going to use my formula for the nth term,0454

an = the first term, times rn - 1.0458

Let's look at what I am given: the first term is 2048: 1, 2, 3, 4, 5...the fifth term is 8; OK.0463

Using this formula: a...I need to find the common ratio, and I can do that because I have the first and last terms, and I have n.0476

1, 2, 3, 4, 5; n = 5, right here; so an is 8; that is equal to the first term, times the common ratio, raised to the power 5 - 1.0485

8 = 2048 times r4; so if I take 8/2048, equals r4, this simplifies to 1/256 = r4.0501

If you think about your roots and your powers, the fourth power is actually plus or minus...0524

I am going to take the fourth power of 1/256; it is actually ± 1/4.0535

And if you multiply this out, you will find that 42 is 16, times 2 is 32, times 2 is 64...0540

Excuse me, 1/4 times 1/4 is 1/16, and continuing on, you will find that the fourth power of 1/4 is 1/256.0551

OK, the important thing to note, though, is that it is not just that the fourth power is 1/256; it is actually plus or minus,0568

because I could take -1/4 times -1/4 times -1/4 times -1/4, and I would also get 1/256.0579

Therefore, with geometric means, you may end up with two sets of answers.0591

All right, so I found my common ratios, which could be ± 1/4; and I have my first term, 2048.0599

So, to find my second term, I just take 2048 times 1/4; that is 5/12.0606

To find my third term, I just take 512, times 1/4; and I am going to get 128.0614

To find the fourth term, I am going to take 128 times 1/4, and I am going to get 32.0626

So, that is one set of answers; I actually have two sets of answers.0635

If r = 1/4, then the missing terms (the geometric means) are 512, 128, and 32.0639

If r is actually equal to -1/4, then the signs will alternate, so what I am going to get is 2048, then -512, then positive 128, then -32.0665

So, I have two possible sets of geometric means.0682

Again, to find the geometric means, I am just going to find the common ratio.0686

And I was able to do that using this formula; I got two answers--the common ratio is either plus or minus 1/4.0691

I took the first term; I multiplied it by 1/4, and then multiplied that by 1/4, and on to get this set of geometric means.0698

I took my other possible solution, r = -1/4, and multiplied 2048 by that to get -512, times -1/4 is 128, times -1/4 is -32.0705

So, that is something to keep in mind: that, with geometric means, you can get two sets of solutions.0718

Example 1: Find the ninth term of the geometric sequence with a fifth term of 80 and a common ratio of 2.0726

The formula for the nth term is first term, times rn - 1, the common ratio to the n - 1 power.0734

Here, n = 9; and I am given a5, and I am already given r.0744

So, let's go ahead and work on this.0755

I have my common ratio of 2, and n is 9, so that is 9 - 1; so a9(which is what I am looking for) equals0760

the first term, times 28; I am stuck.0776

I can't go any farther; I need the first term--I need a1.0780

But there is something else I haven't used yet, and that is the fact that I know a5.0786

In order to go over here and find the first term, I can do that by saying, "OK, let's look at this formula again."0791

I know what a5 is; so let's look at this formula again and use it to solve for the first term.0800

a5 equals the first term, times r; in this case, n will be 5; and that is to the 5 - 1.0809

So, I can substitute in; I know that 80 equals the first term, times r4; and I already do know r, so let's put that in, as well.0818

The first term is 2 to the fourth; 2 times 2 is 4, times 2 is 8, times 2 is 16.0827

So, 24 is actually 16; if I divide both sides by 16, I am going to get 80/16, and the first term is 5.0838

Now, I can go back and finish my problem.0849

I know that a1 is 5; since a1 = 5, let's finish this out.0852

a9 = 5 times 28; when you continue on with powers of 2, we know that 24 is 16.0861

If you continue on up, you are going to find that 28 is 256.0870

So, a9 = 5(256); 5 times 256 is 1280, and that is what we were looking for.0875

All right, again, it looked straightforward, but we had to take a detour, because when we started out0886

using that formula to find the ninth term, we discovered that we got stuck at this step, because we didn't have the first term.0893

But, since they gave us another term and the common ratio, I was able to go back,0902

substitute in 80 here, put in my common ratio of 2, and solve for the first term.0906

Then, I finish out the problem to find that the ninth term is equal to 1280.0914

Find the geometric means: so we need to find the three missing terms.0919

I always look at what I am given first.0923

Well, I am given the first term, and I am given 1, 2, 3, 4, 5...the fifth term.0925

As always, I am going to use my formula here, that the nth term is equal to the first term, times the common ratio raised to the n - 1 power.0934

To find the geometric means, I need the common ratio--what is r?0944

If I have r, I multiply it by 4 to get the second term, then the second term by the common ratio to get the third, and so on.0948

But I don't know r: what I do know are these two things, so I can find r.0957

I have that a5 is 324; and my first term is 4; in this case, n is going to be 5; now I can find the common ratio.0961

This gives me 324 = 4r4; divide both sides by 4--that gives me 81 = r4.0978

The fourth root of 81 is 3; 3 to the fourth power is 81.0988

But there is something I have to remember: the other fourth root of 81 is -3.0995

If I take -3 times -3 times -3 times -3, that is 9, -27, times -3 is also 81.1002

So, I have two possibilities here: r can equal plus or minus 3.1013

I am going to have two sets of results here.1023

Let's let r equal 3; if r equals 3, then I am going to end up with 4 as my first term; I add 3 to that--I am going to get 7.1027

I am going to go ahead and (let's see) add 3 to that...1039

Actually, a correction: I was thinking of arithmetic series; this is a geometric series--I am going to multiply.1051

I need to multiply each term, so 4 times 3 is going to give me 12, times 3 is going to give me 36, times 3 will give me 108, times 3 is 324.1057

So, make sure, when you are working with geometric series, that you are multiplying, not adding.1076

So again, if r = 3, I am going to get geometric means of 12, 36, and 108.1080

If r equals -3, I am going to get 4 times -3 is -12; -12 times -3 is going to give me positive 36, times -3 is -108.1087

They are alternating signs; so there are two possible solutions for the geometric means: 12, 36, 108; or -12, 36, -108.1100

Write an equation for the nth term of the geometric sequence -2, 1/2, and -1/8.1112

The formula for the general term is the first term, times the common ratio raised to the n - 1 power.1123

So, if I am looking for an equation for the nth term here, I am going to need the common ratio.1133

To find the common ratio, I will just take a term and divide it by the previous term.1140

The common ratio...I could take 1/2, and I am going to divide that by -2.1145

Recall that I could just rewrite this, to make it a little clearer, as 1/2 divided by -2--just write it out.1148

And that is the same as multiplying 1/2 by the inverse of -2, and the inverse of -2 is -1/2.1156

So, that is -1/4; the common ratio is -1/4.1171

Now, I can go ahead and write my equation: an = the first term, which is -2, times (-1/4)n - 1.1178

And I only have three terms here; but just writing it in a more general form...n - 1, because it is just asking me for the nth term.1194

-2 times -1/4...a negative times a negative is going to be a positive, so that is just going to be 2/4, or 1/2.1210

Oh, actually, I cannot simplify that any further--correction.1225

I can't simplify that any further, because it is (-1/4)n - 1; we are actually done at this step.1228

We are done right here with the general formula, because I don't have n.1234

All right, so the equation for the nth term is simply going to be an = -2(-1/4)n - 1.1238

So, I could find any term from this geometric sequence, using this equation.1252

All right, write the next three terms of the geometric sequence: -1/3, 1/2, -3/4.1260

In order to find a term, I need to have the common ratio.1268

So, let's find that common ratio by taking 1/2 and dividing it by the previous term, which is -1/3.1272

This is the same as 1/2 divided by -1/3; and remember, I can always rewrite that as 1/2 times the inverse, which is -3, or -3/2.1279

Therefore, r = -3/2; now that I have the common ratio, I can find the next three terms.1290

So, we stopped with the third term--I am looking for the fourth term, the fifth, and the sixth.1297

So, the fourth term is going to be equal to -3/4, times -3/2; this is just going to be 9/8.1306

The fifth term is going to be equal to 9/8, times that common ratio of -3/2.1317

-3/2 times 9/8 is going to give me -27/16: a6 (the sixth term) is going to be -27/16 times -3/2.1326

A negative and a negative is going to give me a positive, and 27 times 3 is actually 81.1343

16 times 2 is 32; and you could leave these as fractions, or you could rewrite them as mixed numbers.1349

I am just going to leave them as they are; so the next three terms are 9/8, -27/16, and 81/32.1357

And I could have looked here and just predicted that the common ratio is negative, because I have these alternating signs: negative, positive, negative.1366

That concludes this lesson on Educator.com, covering geometric sequences; thanks for visiting!1376

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