# Finding the nth term of a quadratic sequence

Look again at the sequence of square numbers:

The diagram shows that:

• the first term is ( )
• the second term is ( )
• the third term is ( )
• the fourth term is ( )

So the term is .

Whenever a sequence has a second difference of , it will be connected to the sequence of square numbers and the term will have something to do with .

Question

What is the term of the sequence , , , , ... ?

The second difference is , so the term has something to do with .

The sequence of square numbers is: , , , , ...

Our sequence is: , , , , ...

Can you see what the difference is?

Each term is higher than the corresponding term in the sequence of square numbers, so the rule for the term is .

Question

What is the term of the sequence , , , , ...?

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

The sequence of square numbers is: , , , , ...

Our sequence is: , , , , ...

Each term in this sequence is less than the corresponding term in the sequence of square numbers, so the rule for the term is .