Chapter 12: Sequences

Sequences

  • An arithmetic or linear sequence is a sequence of numbers where the difference between consecutive terms is a constant
  • In your exam, you might need to work out the nth terms in a sequence
  • Look at this example which shows you how to do it in four steps
    • 1, 5, 9, 13, 17
    • work out zero term -3
    • Work out a formula for the nth term of the sequence
    • Nth terms = difference x n + zero term
    • nth term = 4n - 3
    • Is x in this sequence
    • You can use the nth term to check whether a number is a term in the sequence
    • The value of n in your nth term has to be a positive whole number

Geometric Sequences

  • In a geometric sequence, the ratio between consecutive is constant
    • Here are twp examples of geometric sequences:
    • 3, 9, 27, 81, 243
    • 2, 4, 8, 16, 32

Sequences and equations

  • You can use the nth term or the term-term rule of a sequence to write an equation
  • This sequence has term-to-term rule ‘multiply by 2 then add 4’
    • 11, 25, 53
    • x2 + a x2 +a
    • So 2x11 + a = 25 and a=3
    • You could use this information to find the next term in the sequence

Problem solved

  • Work out what information you need to solve the problem
  • You can’t find the first term until you known that value of k
  • You know two consecutive terms so you can solve an equation to find the value of k

Fibonacci sequence

  • The rule for generating this sequence is ‘add two consecutive terms to get the next term’
    • 2,3,5,8,13,21

Quadratic sequences:

  • If the nth term of a sequence contains an n^2 term and no high er power of n, it is called a quadratic sequence
  • You can write the nth term of a quadratic sequence as:
    • Un = an^2 + bn + c
  • Where a, b, c are numbers and a is not 0
  • You need to be able to find the nth term of a quadratic sequence
  • You can use the golden rule on the right to help

Golden rule

  • The second difference of a quadratic sequence are constant
  • The quadratic sequence with nth term Un =n an^2 + bn+ c has second difference equal to 2a

Working it out

  • Start by writing out the number of coins in each pattern as a number sequence.
  • You are told the sequence is quadratic so you know the second difference will be constant
  • The coefficient of n^2 in the nth term is half of the second difference
  • The second difference are a, so the value of ab is 0.5
  • Once you have worked out the value of a, draw a table
    • You need to compare the values of the terms Un with the quadratic you have
    • This will help you find the rest of the nth term
    • Add a row for Un - an^2
    • This row will form an arithmetic sequence with nth term bn + c
    • The arithmetic sequence has an nth term
    • This is the last past of the nth term of the quadratic sequence
    • Then check you have the write answer