Quadratic Sequence
We are given the following sequence:
-3, 0, 5, 12, 21, ...
The differences between consecutive terms are:
3 , 5 , 7 , 9, ...
Since the differences form an arithmetic sequence, it's a quadratic sequence.
Note the differences of the differences are constant:
2 , 2, 2, ...
The explicit function of the sequence has the form:
[tex]S_n=an^2+bn+c[/tex]
The value of a is the constant difference of differences found divided by 2, thus:
a = 2/2 = 1
a = 1
We still need to find b and c. To do that we form the sequence: Sn - n^2
Subtract from each value of the original sequence, the square of the number of the term:
-3, 0, 5, 12, 21, ..
- 1, 4, 9, 16, 25,...
= -4,-4,-4, -4, -4
This is a constant sequence with the first value = -4 and a common difference of 0. The function of this sequence is: -4
Thus, the explicit function is
[tex]S_n=n^2-4[/tex]
