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SEQUENCES AND SERIES

Home > My Courses > Cambridge Mathematics International AS and A Level – Further Pure Mathematics 1 > SEQUENCES AND SERIES
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Cambridge Mathematics International AS and A Level - Further Pure Mathematics 1

MATRICES
Matrix Structure
  • Video – Matrix Definition (3:48)
Matrix Operations and Definitions
  • Video – Matrix Equality (6:54)
Matrix Algebra
  • Video – Matrix Addition (4:12)
  • Video – Matrix Subtraction (2:27)
  • Video – Scalar Multiples of Matrices (2:09)
  • Video – Zero or Null Matrix (2:18)
  • Video – Negative Matrix (8:26)
Matrix Multiplication
  • Video – Single Row by Single Column (3:41)
  • Video – Single Row by Multiple Columns (6:27)
  • Video – Multiple Rows by Multiple Columns (11:39)
  • Video – Commutative Law (6:48)
  • Video – Power of a Matrix (4:11)
Properties of Matrix Multiplication
  • Topic – Commutative Law of Matrices
  • Topic – Commutative Law with Zero Matrix
  • Topic – Verifying Distributivity of Matrix Multiplication using 2 by 2 Matrices
  • Topic – Associative Law of Matrix Multiplication using 2 by 2 Matrices
  • Topic – Identify Matrix of Multiplication using 2 by 2 Matrices
  • Topic – Square of 2 by 2 Matrices
  • Topic – Cube of 2 by 2 Matrices
  • Topic – Square of 3 by 3 Matrices
  • Topic – Conditions of Square a Matrix
Determinants
  • Video – Understanding Determinant of a Matrix (5:36)
  • Video – Finding Determinant of a Matrix (5:28)
  • Video – Inverse Matrix using Determinant (4:14)
Inverse of a Matrix
  • Video – Finding a Matrix Inverse (10:43)
  • Video – Finding Matrices (8:02)
  • Video – Inverse of an Inverse Matrix (2:48)
  • Video – Inverse of Matrix Multiplications (9:05)
  • Video – Inverse Matrix in a Linear Form (1:55)
SEQUENCES AND SERIES
Sigma Notation for Sums of Arithmetic Sequences
  • Video – Sigma Notation of Arithmetic Series (10:30)
  • Topic – Matching Sigma Notations and Number Series
  • Topic – Arithmetic Series in Sigma Notations
  • Topic – Last Order of Sigma Notations
  • Topic – Constant Series in Sigma Notations
Sigma Notation for Sums of Geometric Sequences
  • Video – Sigma Notation of Geometric Series (6:04)
  • Topic – Listing Geometric Series in Sigma Notations
  • Topic – Expanding Geometric Series in Sigma Notation
  • Topic – Evaluating Geometric Series in Sigma Notation 1
  • Topic – Evaluating Geometric Series in Sigma Notation 2
  • Topic – Evaluating Geometric Series with Coefficients in Sigma Notation
  • Topic – Evaluating Geometric Series using Formula 1
  • Topic – Evaluating Geometric Series using Formula 2
Sum of an Infinite Geometric Series
  • Video – Finding the Limiting Sum (6:39)
  • Video – Finding the First Term and the Common Ratio (3:10)
  • Video – Recurring Decimals (5:23)
  • Video – Recurring Decimal with an Extra Value (3:38)
  • Video – Sum to Infinity or Limiting Sum of Geometric Series (6:41)
  • Video – No Existence of Limiting Sum (1:09)
  • Video – Recurring Decimals involving Limiting Sum (5:56)
  • Video – Application of Limiting Sum (5:12)
  • Video – Limiting Sum involving Success or Failure (8:30)
  • Video – Limiting Sum involving Success Failure or Draw (4:12)
PRINCIPLE OF MATHEMATICAL INDUCTION
Basic Mathematical Induction
  • Video – Introduction to Proof by Mathematical Induction (4:59)
  • Video – Proof by Mathematical Induction 1 (9:09)
  • Video – Proof by Mathematical Induction 2 (5:07)
  • Video – Proof by Mathematical Induction 3 (5:50)
  • Video – Proof by Mathematical Induction 4 (5:06)
  • Video – Proof by Mathematical Induction 5 (4:05)
  • Video – Proof by Mathematical Induction 6 (5:47)
  • Topic – Basic Structure of Mathematical Induction
  • Topic – Sum of Even Numbers by mathematical Induction: Assumption
  • Topic – Sum of Even Numbers by Mathematical Induction: Proof
  • Topic – Summation Proof by Mathematical Induction
  • Topic – Production Formula by Mathematical Induction
  • Topic – Sum of Fractions by Mathematical Induction
  • Topic – Production of Fractions Proof by Mathematical Induction
Consideration of Initial Values
  • Video – Mathematical Induction: Initial Values (8:14)
  • Video – Mathematical Induction: Working with Indices (6:56)
  • Topic – Sum of Odd Numbers by Mathematical Induction: Assumption
  • Topic – Sum of Odd Numbers by Mathematical Induction: Proof
  • Topic – Sum of Terms with Indices by Mathematical Induction
Mathematical Induction with Factorials
  • Video – Mathematical Induction with Factorials 1 (10:16)
  • Video – Mathematical Induction with Factorials 2 (4:53)
  • Topic – Sum of Factorials by Mathematical Induction
Mathematical Induction: Two Initial Values
  • Video – Mathematical Induction: Two Initial Values (14:13)
  • Topic – Two Initial Values in Mathematical Induction
Inequality Proofs by Comparisons
  • Video – Mathematical Induction Inequality: Basic Comparison (6:43)
  • Topic – Inequality proof by Basic Comparison
Inequality Proofs using Differences
  • Video – Mathematical Induction Inequality using Differences (8:31)
  • Topic – Inequality Proof using Differences
Inequality Proofs by Finding Initial Values
  • Video – Mathematical Induction Inequality: Finding Initial Values (9:06)
  • Topic – Finding Initial Values for Proving Inequality
  • Topic – Inequality Proof of Finding Initial Values
Inequality Proofs using Assumptions
  • Video – Mathematical Induction inequality using Assumption (6:19)
  • Topic – Inequality Proof using Assumptions
QUADRATIC EQUATIONS
What are Quadratic Equations?
  • Video – Simple Quadratic Equations (3:31)
  • Video – Basic Quadratic Equations (3:05)
  • Topic – Quadratic Equations in Standard Form 1
  • Topic – Quadratic Equations in Standard Form 2
  • Topic – Quadratic Equations in Standard Form 3
Finding Solutions by Inspecting Graphs
  • Topic – Number of Roots of Quadratic Equations by Inspecting Graphs
  • Topic – Roots of Quadratic Equations by Inspecting Graphs
Forming Quadratic Equations
  • Video – Forming Quadratic Equations (4:17)
  • Video – Form Quadratics involving Fractional Roots (3:09)
  • Video – Finding Quadratic Equations from Solutions (4:56)
  • Video – Theory of Forming Quadratic Equations (5:35)
  • Video – Forming Quadratic Equations: Integral Roots (2:35)
  • Video – Forming Quadratic Equations: Rational Roots (4:46)
  • Video – Forming Quadratic Equations: Irrational Roots (4:15)
Sum and Product of Roots
  • Video – Sums and Products of Roots of Quadratics 1 (3:15)
  • Video – Sums and Products of Roots of Quadratics 2 (6:13)
  • Video – Sums and Products of Roots of Quadratics 3 (6:08)
  • Video – Sums and Products of Roots of Quadratics: Applications 1 (3:31)
  • Video – Sums and Products of Roots of Quadratics: Applications 2 (8:04)
  • Video – Sums and Products of Roots of Quadratics: Applications 3 (6:39)
  • Video – Sums and Products of Roots of Quadratics: Applications 4 (6:30)
  • Video – Sums and Products of Roots involving Graphs (6:46)
  • Video – Sums and Products of Roots involving Circles (6:06)
Substitutions into Quadratics
  • Video – Finding y-values Given x-values from Quadratics (2:42)
  • Video – Quadratics: Substituting Points (1:51)
  • Video – Quadratics: Finding x given y (5:41)
  • Topic – Substitutions into Quadratics
  • Topic – Checking Quadratics
  • Topic – Substitution into Quadratic Function Values: Single Values
  • Topic – Substitution into Quadratic Function Values: Two Values 1
  • Topic – Substitution into Quadratic Function Values: Two Values 2
  • Topic – Substitution into Quadratic Function Values: Quadratic Formula
  • Topic – Substitution into Quadratic Function Values: No Values
Reducible to Quadratics: Polynomials
  • Video – Reducible to Quadratics from Quartic Equations (6:30)
  • Video – Reducible to Quadratics from Non-Monic Quartic Equations (3:29)
  • Video – Reducible to Quadratics from Sextic Equations (2:02)
  • Video – Reducible to Quadratics in Fractions (6:01)
  • Video – Reducible to Quartic Equations in Fractions (6:42)
  • Video – Reducible to Quadratics using Substitutions (3:32)
  • Video – Substitutions to Quartic Equations (2:54)
  • Video – Quadratic Substitutions to Quadratic Equations (3:17)
  • Video – Reducible to Quadratics using Exponents (3:25)
  • Video – Reducible to Quadratics using Logarithms (9:40)
  • Video – Reducible Quadratic Equations of Indices (8:00)
  • Video – Reducible to Quadratics involving Fractions (1:54)
  • Video – Reducible to Quadratics involving Indices (6:15)
  • Video – Reducible to Quadratics involving Surds & Fractions (1:31)
ROOTS AND COEFFICIENTS OF POLYNOMIALS
Roots and Coefficients of Cubic Polynomials
  • Video – Roots & Coefficients of Quadratic Cubic Quartic Polynomials (5:11)
  • Video – Roots & Coefficients of Cubic Polynomials 1 (3:55)
  • Video – Roots & Coefficients of Cubic Polynomials 2 (4:11)
  • Video – Roots & Coefficients of Cubic Polynomials 3 (6:12)
  • Video – Roots & Coefficients of Cubic Polynomials 4 (6:50)
  • Video – Roots & Coefficients of Cubic Polynomials with Product of Roots (9:42)
  • Video – Roots & Coefficients of Cubic Polynomials with Arithmetic Progression (6:00)
  • Video – Roots & Coefficients of Cubic Polynomials with Sum of Roots (4:17)
  • Video – Solving Cubic Polynomials involving Roots and Coefficients (1:52)
GRAPHS
Rational Functions
  • Video – Domain and Range of Rational Expressions 1 (3:53)
  • Video – Domain and Range of Rational Expressions 2 (3:29)
  • Video – Domain and Range of Rational Expressions 3 (5:42)
Graphs of Rational Functions
  • Video – Drawing Graph of Rational Function: Step 1 (1:25)
  • Video – Drawing Graph of Rational Function: Step 2 (2:04)
  • Video – Drawing Graph of Rational Function: Step 3 (0:28)
  • Video – Drawing Graph of Rational Function: Step 4 (0:28)
  • Video – Drawing Graph of Rational Function: Step 5 (2:59)
  • Video – Drawing Graph of Rational Function: Step 6 (4:17)
  • Video – Drawing Graph of Rational Function: Step 7 (1:11)
  • Video – Drawing Graph of Rational Function: Step 8 (2:28)
  • Video – Graphs of Rectangular Hyperbola (2:17)
  • Video – Graphs of Hyperbola involving Oblique Asymptotes (2:45)
Curves of Rational Functions in Quadratics by Linear Form
  • Video – Understanding Curve Sketching involving Asymptotes (5:00)
  • Video – Finding Oblique Asymptotes Vertical Asymptotes (4:06)
  • Video – Turning Points of Hyperbola (3:42)
  • Video – Nature of Turning Points of Hyperbola (7:02)
  • Video – Curve Sketching of Hyperbola involving Asymptotes (0:47)
  • Video – Understanding Curve Sketching involving Symmetry (1:24)
  • Video – Even Functions Odd Functions (1:49)
  • Video – Differentiation using Quotient Rule (2:29)
  • Video – Turning Points of Rational Functions (1:17)
  • Video – Nature of Turning Points of Rational Functions (3:03)
  • Video – x-Intercepts of Rational Functions (0:58)
  • Video – Vertical Asymptotes in Rational Functions (1:53)
  • Video – Horizontal Asymptotes in Rational Functions (4:52)
  • Video – Curve Sketching of Rational Functions involving Symmetry (1:53)
Further Rational Functions
  • Topic – Oblique Asymptotes
  • Topic – Correct Asymptotes
  • Topic – Rational Graphs involving Oblique Asymptotes
Absolute Value (Modulus) Graphs
  • Video – Basic Theory of Absolute Value Graphs (3:30)
  • Video – Sketching Absolute Value Graphs (4:41)
  • Video – Transformation of Absolute Value Graphs: Up Down (3:09)
  • Video – Transformation of Absolute Value Graphs: Left Right (5:01)
  • Video – Absolute Value Graphs Regarding Steepness (3:47)
  • Video – Absolute Value Graphs involving Quadratics (5:28)
  • Topic – Translating Absolute Value (Modulus) Graphs 1
  • Topic – Translating Absolute Value (Modulus) Graphs 2
  • Topic – Translating Absolute Value (Modulus) Graphs 3
  • Topic – Translating Absolute Value (Modulus) Graphs 4
  • Topic – Translating Absolute Value (Modulus) Graphs 5
  • Topic – Translating Absolute Value (Modulus) Graphs 6
  • Topic – Vertices of Absolute Value (Modulus) Graphs
  • Topic – x-intercepts of Absolute Value (Modulus) Graphs
  • Topic – y-intercepts of Absolute Value (Modulus) Graphs
Further Graphs of Absolute Value (Modulus) Functions
  • Video – Absolute Value of Sine Graphs (1:26)
  • Video – Graphing Absolute Value Functions (4:23)
  • Video – Transformation of Absolute Value Graphs (3:29)
  • Video – Graphs involving Absolute Value (1:40)
  • Video – Understanding Absolute Value Graphs (7:37)
  • Video – Absolute Graphs involving Quadratics (3:59)
  • Video – Absolute Value Graphs Given Initial Graphs 1 (5:30)
  • Video – Absolute Value Graphs Given Initial Graphs 2 (7:11)
  • Video – Absolute Value Graphs involving Natural Logarithmic Graphs (1:48)
  • Video – Absolute Value Graphs involving Sine Trigonometric Graphs (2:26)
Further Graphs of Square Root Functions
  • Video – Graphs of Square Root Functions (4:27)
  • Video – Graphs of Square Root Functions Given Initial Graphs (4:24)
  • Video – Odd Powers of Square Root Graphs (0:35)
Reciprocal Functions
  • Video – Reciprocal Graphs (5:31)
  • Video – Reciprocal Graphs of Irrational Functions (1:37)
  • Video – Reciprocal Graphs of Modulus Functions (0:48)
  • Video – Reciprocal Graphs of Natural Exponential Functions (1:01)
  • Video – Reciprocal Graphs of Natural Logarithmic Functions (0:30)
  • Video – Reciprocal Graphs of Inverse Tangent Functions (1:45)
  • Video – Reciprocal Graphs of Given Initial Graphs (2:52)
  • Video – Reciprocal Graphs of Parabola (5:11)
Graphs of Reciprocal Trigonometric Functions
  • Video – Reciprocal Trigonometric Functions: Cosecant Graphs (5:26)
  • Video – Reciprocal Trigonometric Functions: Secant Graphs (3:24)
  • Video – Reciprocal Trigonometric Functions: Cotangent Graphs (3:39)
  • Video – Graphs of Reciprocal Trigonometric Functions (7:24)
VECTORS
Applications of a Line in a Plane
  • Topic – Initial Positions from Vector Equations
  • Topic – Finding Speed of a Particle using its Vector Equation
  • Topic – New Vector Equations for New Velocities
  • Topic – Position of Vectors given Certain Times
  • Topic – 2-Dimensional Velocity Vectors given New Speeds
  • Topic – Time for Vector Equations
  • Topic – Speed of Particles described as 2-Dimensional Vectors
  • Topic – 3-Dimensional Velocity Vectors given New Speeds
  • Topic – Shortest Distance using Vectors
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