Subjects

• A sequence is an ordered list of numbers or terms that follow a particular pattern
• Sequence


  A function that maps natural numbers (including zero) to a set of numbers or terms
• Types of sequences


  • Arithmetic sequences
  • Geometric sequences
  • Other types of sequences
• Arithmetic sequences


  • Have a common difference between consecutive terms
• Arithmetic sequence


  • 2, 5, 8, 11, ...
• Geometric sequences


  • Have a common ratio between consecutive terms
• Geometric sequence


  • 3, 6, 12, 24, ...
• Other types of sequences

  • 1, 4, 9, 16, 25, ...
• Characteristics of sequences


  • Order
  • Pattern
  • Term position
  • Common difference or ratio
• The study of sequences dates back to ancient civilizations
• Arithmetic sequence example


  • 2, 5, 8, 11, ...
• Finding the 10th term of the arithmetic sequence

  1. Use the formula an = a1 + (n - 1)d

  2. Plug in the values a1 = 2, n = 10, d = 3

  3. a10 = 2 + (10 - 1)3 = 29
• Geometric sequence example


  • 3, 6, 12, 24, ...
• Finding the 6th term of the geometric sequence


  1. Use the formula an = a1 * r^(n-1)

  2. Plug in the values a1 = 3, n = 6, r = 2

  3. a6 = 3 * 2^(6-1) = 96
• Recursive definition


  Each term is defined in relation to the previous terms in the sequence
• Recursive definition example


  • Fibonacci sequence: F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n ≥ 2
• Explicit definition


  The nth term is defined directly in terms of n or an equation
• Explicit definition example


  • Arithmetic sequence: an = a1 + (n-1)d
• Finding the nth term


  Determine the value of a specific term in a sequence based on its position
• Polynomials play a significant role in mathematics, science, and engineering
• Polynomial


  Algebraic expressions that consist of variables, coefficients, and exponents
• Types of polynomials


  • Constant polynomials
  • Linear polynomials
  • Quadratic polynomials
  • Cubic polynomials
  • Higher degree polynomials
• Characteristics of polynomials


  • Degree
  • Coefficients
  • Constant term
  • Leading coefficient
  • Zero polynomial
• Polynomials have a rich history dating back to ancient civilizations
• Polynomial example


  • P(x) = 3x³ - 2x² + 5x - 1
• Polynomial equation example


  • Solve the equation P(x) = 0, where P(x) = x² + 3x - 4
• Polynomial functions are fundamental mathematical functions that involve variables raised to non-negative integer powers and combined using addition, subtraction, and multiplication
• Polynomial function


  Functions that can be expressed as f(x) = anx^n + an-1x^(n-1) + ... + a1x + a0
• Types of polynomial functions

  • Constant functions
  • Linear functions
  • Quadratic functions
  • Cubic functions
  • Higher degree polynomial functions
• Characteristics of polynomial functions

  • Degree
  • Leading coefficient
  • Coefficients
  • Zeros and roots
• The study of polynomial functions has a rich history dating back to ancient civilizations
• Constant function example


  • f(x) = 5
• Linear function example


  • f(x) = 3x + 2
• Quadratic function example


  • f(x) = x² - 4x + 3
• Roots or zeroes of a polynomial function


  The values of x for which the function evaluates to zero
• Finding the roots of the quadratic function f(x) = x² - 4x + 3


  • Set f(x) = 0 and solve the quadratic equation
  • The roots are x = 1 and x = 3
• Multiplicity of a root


  • The number of times a root appears as a solution to the polynomial equation
• Polynomial division is an important operation for working with polynomial functions
• Quadratic function


  Degree 2 polynomial function with a leading coefficient of 1, a coefficient of -4, and a constant term of 3
• Roots or zeroes of a polynomial function


  Values of x for which the function evaluates to zero