Quadratic Sequences

Quadratic Sequences

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 nth term formula n² are the square numbers 

  • 1, 4, 9, 16, 25, 36, 49, ...

    • From 1², 2², 3², 4², ...

• Some sequences are just the square numbers but with a different starting number

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

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

  • 0, 1, 4, 9, ... has the formula (n - 1)²

• Finding the nth 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 the above

  • For example, doubling then adding 1

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

  • For example, subtracting 2 from square numbers starting with 4

    • 2, 7, 14, 23, ... has the formula (n + 1)² - 2

How do I find the n^th term formula for any 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 part of the formula

  • 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 term

  • e.g.  an² + bn + c = 2n² + 3n − 4