Quadratic Sequences (Cambridge (CIE) IGCSE International 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, ...

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

• 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, ...

• Finding the n th term formula comes from comparing sequences to the square numbers

  • 2, 5, 10, 17, 26, 37, 50, ... has the formula n² + 1

    • Each term is 1 more than the square numbers

  • 2, 8, 18, 32, 50, 72, 98, ... has the formula 2n²

    • Each term is double a square number

• It may be a simple combination of both

  • For example, doubling then adding 1

    • 3, 9, 19, 33, 51, 73, 99, ... has the formula 2n² + 1

• Some sequences are just the square numbers but starting later

  • 16, 25, 36, 49, ... has the formula (n + 3)²

    • Substitute in n = 1, n = 2, n = 3 to see why

• You can also use second differences to help find the n th term

  • For the simple quadratic sequence an² + b

    • a  is half of the second difference

  • E.g. for the sequence 3, 9, 19, 33, 51, ...

    • The second difference is 4, so a = × 4, a = 2

    • Compare the sequence 2n², (2, 8, 18, 32, 50, ...) to the original sequence

    • The original sequence has the n th term rule 2n² + 1