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# Finding the nth term

Look again at the sequence of square numbers:

The diagram shows that:

• the first term is 1 (12)

• the second term is 4 (22)

• the third term is 9 (32)

• the fourth term is 16 (42)

So the nth term is n2

Whenever a sequence has a second difference of 2, it will be connected to the sequence of square numbers and the nth term will have something to do with n2.

Question

What is the nth term of the sequence 3, 6, 11, 18, 27, ... ?

The second difference is 2, so the nth term has something to do with n2.

The sequence of square numbers is: 1, 4, 9, 16, 25, ...
Our sequence is: 3, 6, 11, 18, 27, ...

Can you see the difference?

Each term is 2 bigger than the corresponding term in the sequence of square numbers, so the rule for the nth term is n2 + 2.

Question

What is the nth term of the sequence 0, 3, 8, 15, 24, ... ?

n2 - 1

The second differences are 2, so the formula has something to do with n2.

The sequence of square numbers is: 1, 4, 9, 16, 25, ...
Our sequence is: 0, 3, 8, 15, 24, ...

Each term in our sequence is 1 less than the corresponding term in the sequence of square numbers, so the rule for the nth term is n2 - 1.

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