Finding the nth term of a quadratic sequence

Look again at the sequence of square numbers:

Quadratic sequence diagram

The diagram shows that:

  • the first term is {1} ( {1}^{2})
  • the second term is {4} ( {2}^{2})
  • the third term is {9} ( {3}^{2})
  • the fourth term is {16} ( {4}^{2})

So the n^{th} term is n^2.

Whenever a sequence has a second difference of {2}, it will be connected to the sequence of square numbers and the n^{th} term will have something to do with n^2.

Question

What is the {n}^{th} term of the sequence 3, 6, 11, 18, 27 ... ?

Quadratic sequence diagram
Quadratic sequence diagram

The second difference is 2, so the {n}^{th} term has something to do with n^2.

The sequence of square numbers is: 1, 4, 9, 16, 25 ...

Our sequence is: 3, 6, 11, 18, 27 ...

Can you see what the difference is?

Each term is 2 higher than the corresponding term in the sequence of square numbers, so the rule for the {n}^{th} term is {n}^{2} + 2.

Question

What is the {n}^{th} term of the sequence 0, 3, 8, 15, 24 ...?

Quadratic sequence diagram

{n}^{2}-{1}

The second differences are 2, so the formula has something to do with n^2.

Quadratic sequence diagram

The sequence of square numbers is: 1, 4, 9, 16, 25 ...

Our sequence is: 0, 3, 8, 15, 24 ...

Each term in this sequence is 1 less than the corresponding term in the sequence of square numbers, so the rule for the {n}^{th} term is {n}^{2}-{1}.