INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Infinite Geometric Series

Dr. Carleen Eaton

Dr. Carleen Eaton

Infinite Geometric Series

Slide Duration:

Table of Contents

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
Addition Property
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
Addition Property
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
Quadrants
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
Shaded Region
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
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
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
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
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
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
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
Addition and Subtraction
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
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
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
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
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
Examples: Adding Monomials
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
Leading Coefficient
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
Quadratic Form
8:22
Example: Quadratic Form
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
Radicand
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
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
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
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
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
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
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
Adding and Subtracting
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
Example 4: Solve Log Inequality
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
Example: Log Inequality
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
Example 4: Log Inequality
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
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
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
Radioactive Decay
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
Radius
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
Example 3: Radius
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
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
Loading...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Algebra 2
Bookmark & Share Embed

Share this knowledge with your friends!

Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
  ×
  • - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

  • Discussion

  • Answer Engine

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Please login to ask a question and view discussion.

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
Get immediate access to our entire library.

Sign up for Educator.com

Membership Overview

  • Unlimited access to our entire library of courses.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lesson files for programming and software training practice.
  • Track your course viewing progress.
  • Download lecture slides for taking notes.
  • Learn at your own pace... anytime, anywhere!

Infinite Geometric Series

  • Remember that an infinite geometric series converges only if |r| < 1. If you are given a series to evaluate, first check the value of r. If it does not satisfy this condition, the series does not have a sum.
  • To convert a repeating decimal to a fraction, let r = 10-n, where n = number of digits in the repeating pattern of the decimal. Let a1 = the fraction which contains the pattern written once in the numerator and 10n in the denominator.

Infinite Geometric Series

Find the sum 1 − [4/5] + [16/25] − [64/125]
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [( − [4/5])/1] = − [4/5]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [1/(1 − ( − [4/5] ))] =
  • [1/(1 + [4/5])] = [1/([9/5])] = [5/9]
S = [5/9]
Find the sum [5/3] − [5/12] + [5/48] − [5/192]
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [( − [5/12])/([5/3])] = − [5/12] ÷[5/3] = − [5/12]*[3/5] = − [3/12] = − [1/4]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [([5/3])/(1 − ( − [1/4] ))] =
  • [([5/3])/(1 − ( − [1/4] ))] = [([5/3])/(1 + [1/4])] = [([5/3])/([5/4])]
  • [([5/3])/([5/4])] = [5/3]*[4/5] = [4/3]
S = [4/3]
Find the sum 4.4 + 3.52 + 2.816 + 2.2528
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [3.52/4.4] = 0.8
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [4.4/(1 − ( 0.8 ))] =
  • [4.4/(1 − ( 0.8 ))] = [4.4/0.2] = 22
S = 22
Find the sum 9.3 + 1.86 + 0.372 + 0.0744...
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [1.86/9.3] = 0.2
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [9.3/(1 − ( 0.2 ))] =
  • [9.3/(1 − ( 0.2 ))] = [9.3/0.8] = 11.625
S = 11.625
Find the sum 5 + [5/2] + [5/4] + [5/8]...
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [([5/2])/5] = [5/2] ÷[5/1] = [5/2]*[1/5] = [1/2]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [5/(1 − ( [1/2] ))] =
  • [5/(1 − ( [1/2] ))] = [5/([1/2])] = 10
S = 10
Find the sum [1/2] + [1/10] + [1/50] + [1/250]
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [([1/10])/([1/2])] = [1/10] ÷[1/2] = [1/10]*[2/1] = [1/5]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [([1/2])/(1 − ( [1/5] ))] =
  • [([1/2])/(1 − ( [1/5] ))] = [([1/2])/([4/5])] =
  • [([1/2])/([4/5])] = [1/2]*[5/4] = [5/8]
S = [5/8]
Find the sum 6 − 2 + [2/3] − [2/9]
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [( − 2)/6] = − [1/3]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [6/(1 − ( − [1/3] ))] =
  • [6/(1 − ( − [1/3] ))] = [6/([4/3])] =
  • [6/([4/3])] = 6*[3/4] = [18/4] = [9/2]
S = [9/2]
Find the sum − 2 − [2/5] − [2/25] − [2/125]...
  • Step 1:Find the common ratio r, if r is less than 1, then you may find the sum using the formula S = [(a1)/(1 − r)]
  • r = [(a2)/(a1)] = [( − [2/5])/( − 2)] = − [2/5]* − [1/2] = [1/5]
  • Step 2) Since r is < 1, you may find the sum using the formula
  • S = [(a1)/(1 − r)]
  • S = [(a1)/(1 − r)] = [( − 2)/(1 − ( [1/5] ))] =
  • [( − 2)/(1 − ( [1/5] ))] = [( − 2)/([4/5])] =
  • [( − 2)/([4/5])] = − 2*[5/4] = − [5/2]
S = − [5/2]
Write as a fraction 0.25252525....
  • Step 1 - Write the decimal as a geometric series
  • 0.25252525... = 0.25 + 0.0025 + 0.000025 + 0.000025...
  • Step 2 - Find the common ratio r
  • r = [(a2)/(a1)] = [0.0025/0.25] = 0.01
  • Step 3 - Use the formula Sn = [(a1)/(1 − r)]
  • Sn = [(a1)/(1 − r)] = [0.25/(1 − 0.01)] = [0.25/0.99]
  • Move the decimal two times over
  • [0.25/0.99] = [25/99]
0.25252525... = [25/99]
Write as a fraction 0.15151515....
  • Step 1 - Write the decimal as a geometric series
  • 0.25252525... = 0.15 + 0.0015 + 0.000015 + 0.000015...
  • Step 2 - Find the common ratio r
  • r = [(a2)/(a1)] = [0.0015/0.15] = 0.01
  • Step 3 - Use the formula Sn = [(a1)/(1 − r)]
  • Sn = [(a1)/(1 − r)] = [0.15/(1 − 0.01)] = [0.15/0.99]
  • Move the decimal two times over
  • [0.15/0.99] = [15/99]
0.15151515... = [15/99] = [5/33]

*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.

Answer

Infinite Geometric Series

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
  • What are Infinite Geometric Series 0:10
    • Example: Finite
    • Example: Infinite
    • Partial Sums
    • Formula
  • Sum of an Infinite Geometric Series 2:39
    • Convergent Series
    • Example: Sum of Convergent Series
  • Sigma Notation 7:31
    • Example: Sigma
  • Repeating Decimals 8:42
    • Example: Repeating Decimal
  • 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

Transcription: Infinite Geometric Series

Welcome to Educator.com.0000

Today, we continue on our discussion of sequences and series with infinite geometric series.0002

What are infinite geometric series? Well, this is a type of series in which there is an infinite number of terms.0010

Earlier on, I mentioned that, for either arithmetic or geometric series, you may have a limited number of terms,0018

which is a finite series; or it may go on indefinitely, which is an infinite series.0024

For example, the geometric series 1 + 1/4 + 1/16 + 1/64 is a geometric series with 4 terms; and it has a common ratio r = 1/4.0030

This is a finite series: it has a limited number of terms.0047

Consider another geometric series: 3 + 6 + 12 + 24, and then, when you see the ellipses (the three dots),0052

it indicates that it goes on indefinitely; so this is an infinite series.0063

The sums, sn, are called partial sums of the infinite series.0070

For example, I may decide that I want to find the first 7 terms of this series, or the first 5 terms of an infinite geometric series.0074

That would just be a partial sum of this series.0084

And we will talk, in a minute, about special cases, when you actually can find the sum overall of an infinite geometric series.0087

So, recall from the previous lesson the formula for the sum of a geometric series,0097

sn = the first term, times (1 - rn), divided by (1 - r).0103

So, if I were to try to find the first seven terms of this series, s7, I could use this formula.0114

I have the first term; I need to find r; recall that I can find r by taking a term and dividing it by the one that goes just before.0120

So here, r = 2; so this is 3(1 - 2n), so that is 7; and then, divided by 1 - 2.0130

So here, I found the partial sum for this infinite geometric series.0149

You can actually find the sum of an infinite geometric series (not just the partial sum) in some cases.0160

And I know that this sounds counterintuitive--how can you find the sum of something that goes on forever?0166

But if you look, you can see why.0172

All right, first of all, it is very important to know that this is limited to cases in which the series is convergent.0178

And the word "convergent" indicates that the sum converges toward a particular number.0184

A series is convergent if and only if the absolute value of r is less than 1.0191

So, if you are working with a geometric series in which the value of the common ratio is either greater than -1 or less than 1,0196

such as 3/4, for example--if r is 3/4 (the absolute value of that is 3/4), or if r is -1/2 (I take the absolute value of that--it would be 1/2),0205

both of these are convergent; I could actually find the sum of those.0219

Consider the series 1/3 + 1/9 + 1/27, going on indefinitely.0222

Remember, to find r (let's go up here and find r), we are going to take 1/9, divided by 1/3.0234

This is the same as 1/9 times 3, which equals 3/9, which equals 1/3; therefore, r = 1/3.0241

If I am asked to find the sum of this, I go ahead and use this formula, x equals the first term, which is 1/3, divided by 1 - 1/3, equals (1/3)/(2/3).0258

1/3 divided by 2/3 is the same as 1/3 times 3/2, or 3/6.0273

The sum of this infinite geometric series is 1/2.0283

Let's look at this another way: just go ahead and add up some of the terms and see what happens.0288

s1 for this term is 1/3; that is all you have: 1/3.0297

So, s2 would be adding 1/3 + 1/9; 1/3 + 1/9 equals...giving this a common denominator,0304

I multiply both the numerator and the denominator by 3 to get 3/9 + 1/9 is 4/9; that is s2.0313

s3 = 1/3 + 1/9 + 1/27; well, I know that these two are equal to 4/9, so that is 4/9 + 1/27.0322

So again, I need to get a common denominator; and if you work that out, you will find that s3 is 13/27.0333

s4...I would continue on: 1/3 + 1/9 + 1/27 + 1/81...and if you figure that out, it becomes 40/81.0343

I won't work out the rest of these right here; but I will just tell you that s5 is 121/243.0360

s6 is 364/729; and s7 is 1093/2187.0370

Let's look at the pattern here: it started out as 1/3; then it became 4/9, 13/27, 40/81, 121/243, 364/729, 1093/2187.0383

What is happening is: this sum is converging upon 1/2.0400

Another way that we say this is that the limit is 1/2; and that is terminology you will hear later on in higher math courses.0414

But for right now, just be aware that you can only find the sum of an infinite geometric series if it is convergent,0422

meaning that, as you take more and more terms and add them to the series, add them, and get their sum,0428

you will see that the sum of the series is converging upon a particular number.0435

And so, we just use this formula as a great shortcut to find the sum.0440

Sigma notation is something we discussed earlier on.0452

Go back and look at the lectures on geometric sequences and geometric series, if any of these concepts are new.0456

But sigma notation--you will recall that the Greek letter sigma means sum, and we used it with other geometric series.0462

For an infinite series, you are going to have something like this.0470

We have our lower index, i = 1; and this is the series as i goes from 1 to infinity.0476

The difference here is that, instead of stopping at a specific value up here, it goes on through infinity.0484

And again, we have the formula for the sequence right here.0490

Another example, or a specific example, would be a series, again, where i goes from 1 to infinity,0494

but we have a formula over here, 1/4 times 1/2, raised to the n - 1 power.0501

So, what you could do, then, is put 1 in here and find your first term, a1.0507

Then, put 2 here; find your second term; and go on infinitely, because of the type of series that this is.0513

We can use the concepts that we just learned to actually write a repeating decimal as a fraction.0523

The sum formula I just described can be used in this way.0529

A repeating decimal would be something like this: .44444...and it just goes on and on that way.0533

What we do is rewrite this as a geometric series.0542

1: First step--how do you do that?0547

Well, look at what this really means: it really means 0.4 + 0.04 + 0.004 + 0.0004, and so on.0556

So, I rewrote this, but just as a series; and it is an infinite geometric series.0573

Then, find the sum of the series: recall that you can only find the sum of an infinite geometric series0580

if the absolute value of r is less than 1; this is required, or you can't find the sum.0590

Well, let's show here that we are OK, because the absolute value of r is actually less than 1.0604

So, in order to find the common ratio, r, I am going to take .04, and I am going to divide it by the term before it, which is .4.0609

So, I move the decimal over one place; that is going to give me .4/4, which is .1; so the common ratio, r, equals .1.0618

.1 is less than 1, so I am fine: I can use the sum formula, the first term divided by (1 - r).0626

So, the sum equals the first term, divided by (1 - .1), equals .4/.9.0635

Dividing that, move the decimal over; you can get rid of that decimal; that gives me 4/9.0645

So, I found that the sum of this...0652

I started out; I have this decimal that is a repeating decimal (it goes on forever); but I saw that I could rewrite this as a series.0657

So, this, therefore, is equivalent to the sum of the series.0665

My next step was to figure out what the sum of this series is, and it is actually 4/9.0670

Therefore, .444 repeating can be rewritten as 4/9; those are equivalent.0674

Recall that you can have a repeating decimal that doesn't just have one number repeat and repeat; it could be multiple numbers.0683

It could be .383838 repeating, and you could do the same thing.0690

You could rewrite this as .38 + .0038 + .000038, and so on; and then repeat as above, by finding the sum of the series.0698

So, this could also be used for repeating decimals where there is a longer repeat.0717

Also recall that we can write these as .4 with a bar over it, or .38 with a bar over it--that is just a different notation.0722

All right, let's find the sum of this infinite geometric series.0736

First, though, I am going to verify that I actually can find the sum of this by figuring out what r is.0741

So, I am going to take 24, the common ratio, divided by 32; one term divided by the previous one gives me the common ratio.0746

This simplifies out to 3/4: 3/4 is less than 1, so yes, I can find the sum, using this formula.0753

The formula: I need to use the first term: 32/(1 - 3/4) = 32/(1/4).0769

We can rewrite this as 32 times 4; and 32 times 4...4 times 2 is 8; 4 times 3 is 120; so that gives me 128.0778

The sum of this infinite geometric series is 128, and I was able to find that using this formula, because I had an absolute value of r that is less than 1.0791

Write as a fraction: recall that this notation means the same thing as .36363636, and so on.0806

The first step is to write this as a geometric series, so I am going to rewrite this as .36 + .0036 + .000036, and so on.0821

And I am going to use my formula for the sum of an infinite geometric series, which is the first term, divided by (1 - r).0844

What is r? Well, as usual, I can find the common ratio, r, by taking a term, .0036, and dividing by the previous term, .36.0853

Move the decimal over two places to get .36/36; and that is going to give me .01.0864

.01 is less than 1, so I can find the sum.0871

Take the first term; divide it by (1 - .01); this gives me .36/.99.0875

I can move the decimal over two places, to give me 36/99; simplify that, because there is a common factor of 3; this is actually 12/33.0885

I can see, again, that I have another common factor of 3; so this is going to give me 4/11.0896

Therefore, this 0.36 repeating decimal can be rewritten as 4/11; so I wrote this repeating decimal as a fraction.0901

Find the sum for this infinite geometric series: before I proceed, I check that the absolute value of the common ratio is actually less than 1.0918

10/12...to find the common ratio, divide that by the term that came just before, which is 5/4.0927

And this is going to give me 10/12 times 4/5 equals 40/60; so this is going to give me 2/3.0933

And since 2/3 is less than 1, I can find the sum of this series using my formula for the sum of an infinite geometric series.0949

OK, the first term is 5/4; I am dividing that by 1 - 2/3 to get 5/4 divided by 1/3.0961

This is the same as 5/4; and then I just take the inverse of 1/3, and 5/4 times the inverse of 1/3; that is 3, so this is going to give me 15/4.0977

And you can keep this as an improper fraction, or write it as a mixed number.0989

The sum of this infinite geometric series is 15/4.0993

Again, I was able to find that because the common ratio had an absolute value that was less than 1.0999

Write as a fraction: we have another decimal that is repeating--goes on infinitely: 0.99999, and so on.1009

Start out by rewriting this as a geometric series.1021

I have 0.9 + 0.09 + 0.009, and so on; it goes on infinitely.1027

Find the common ratio, r: r =...I am going to take .09, and divide that by .9.1041

Move the decimal over one place to get .9/9; therefore, the common ratio is .1.1048

Next, I use my formula for the sum of an infinite geometric series, which is the first term, divided by 1 - r.1055

And this is going to give me the first term, .9, divided by 1 - .1; that is .9/.9; this is simply 1.1063

So, this asked me to write it as a fraction; and .999 = 1, or you could say 1/1.1081

This is still a fraction, because you could just write it as 1/1.1090

And it is counterintuitive, if you think ".999 repeating is actually 1"; it doesn't seem like it is, but it is actually correct.1093

That concludes this lesson of Educator.com on infinite geometric series; thanks for visiting!1105

Related Books

Related Books

Educator®

Please sign in to participate in this lecture discussion.

Resetting Your Password?
or
Still having trouble? Contact us
OR

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
 Get immediate access to our entire library.

Membership Overview

  • Available 24/7. Unlimited Access to Our Entire Library.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lecture slides for taking notes.
  • Track your course viewing progress.
  • Accessible anytime, anywhere with our Android and iOS apps.