Quadratic Sequences (Edexcel GCSE Maths)

Quadratic Sequences

What is a quadratic sequence?

• A quadratic sequence has an n th term formula that involves n²

• The second differences are constant (the same)

  • These are the differences between the first differences

  • For example,   3, 9, 19, 33, 51, …
    1st Differences: 6, 10, 14, 18, ...

    2nd Differences:  4,   4,   4, ...

• The sequence with the n th term formula n² are the square numbers 

  • 1, 4, 9, 16, 25, 36, 49, ...

    • From 1², 2², 3², 4², ...

How do I find the n^th term formula for a simple quadratic sequence?

• STEP 1
  Work out the sequences of first and second differences

  • e.g. for the sequence 1, 10, 23, 40, 61

• STEP 2
  Divide the second difference by 2 to find the coefficient of n²

  • e.g.  a = 4 ÷ 2 = 2

• STEP 3
  Write out the first three or four terms of an² and subtract the terms from the corresponding terms of the given sequence

  • e.g. for the sequence 1, 10, 23, 40, 61

• STEP 4
  Work out the nth term of these differences to find the bn + c

  • e.g. the nth term of -1, 2, 5, 8, ... is  bn + c = 3n − 4

• STEP 5
  Find an² + bn + c by adding together this linear nth term to an²

  • e.g.  an² + bn + c = 2n² + 3n − 4