Look again at the sequence of square numbers:
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}.
What is the n^{th} term of the sequence 3, 6, 11, 18, 27, ... ?
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 the difference?
Each term is 2 bigger than the corresponding term in the sequence of square numbers, so the rule for the n^{th} term is n^{2} + 2.
What is the n^{th} 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 n^{th} term is n^{2} - 1.
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