Summation of series

• ∑ notation (revision)
• Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
  • Harder examples using these results
• Sum of the squares of the first n natural numbers ∑r²
• Sum of the cubes of the first n natural numbers ∑r³
• Using known formulae to sum more complex series

Exam Questions

Method of Differences

• Method of differences
• Example using factorials

Proof by mathematical induction

Series

• Proof of the sum of the series ∑r
• Proof of the sum of the series ∑r²
• Proof of the sum of the series ∑r³
• Further examples
  • Example 1
  • Example 2

Divisibility and Multiple Tests

• Proof that an expression is divisible by a certain integer (power type)
• Proof that an expression is divisible by a certain integer (non-power type)

Recurrence Relations

• Proof of the general term from a recurrence relationship (example 1)
• Proof of the general term from a recurrence relationship (example 2)

Matrices

• Proof of matrix multiplication statements (example 1)
• Proof of matrix multiplication statements (example 2)

Exam Questions

Roots of polynomial equations

• Relationships between the roots and coefficients of a quadratic equation
• Relationships between the roots and coefficients of a cubic equation
• Relationships between the roots and coefficients of a quartic equation

Complex numbers

• Real and imaginary numbers
• Addition, Subtraction and Multiplying complex numbers and simplifying powers of i
• Complex conjugates
• Division of a complex number by a complex number
• Argand diagram
• Modulus and argument of a complex number
• Mod-Arg form of a complex number
• Solving problems with complex numbers
• Square roots of a complex number
• Solving quadratic equations with complex roots (complex conjugate pairs)
• Solving cubic equations
• Solving quartic equations

Exam Questions

Matrices

• Introduction and dimension of a matrix
• Addition and subtraction and multiplying a matrix by a scalar
• Matrix multiplication
• Identity and Inverse of a 2x2 matrix
  • Example on singular matrices
  • Example on solving a matrix equation
  • Example on proving a matrix identity

Applications

• Simultaneous Equations
  • Solving simultaneous equations
• Linear Transformations
  • Rotations about the origin of:
    • 90^o | 180^o | 270^o | θ^o
  • Reflections in the:
    • x-axis | y-axis | y = x | y = -x
  • Enlargement, scale factor k, centre (0,0) where:
    • | k>0 | k<0
  • Transformations Test:
    • How well do you know your transformations? : Take the Test
  • Combinations of Transformations:
    • Combinations of transformations
  • Inverse Matrices:
    • Inverse matrices to reverse linear transformations
  • Area Scale Factor:
    • Determinant as the area scale factor of a transformation
    • Proof

Exam Questions