How do you find the nth term formula for a sequence with non-constant difference?
Take the sequence;
9, 12, 19, 30, ...
(1) The first step is always to look at difference between the terms;
9, 12, 19, 30, ...
+3, +7, +11, ...
+4, +4, ...
We can see the difference is not constant, (2) so we looked at the change in the difference each term.
This gives a constant change in the difference of an extra +4 each term. The fact that we needed to take 2 turns to find the constant difference means we are dealing with a quadratic sequence.
(3) Furthermore, because the difference is +4, we are dealing with a 2n2 sequence.
If the change in the difference is (a) then the nth term follows a (1/2a)n2 pattern.
(4) Now we can rewrite the sequence as follows;
n n2 2n2
9 1 1 2
12 2 4 8
19 3 9 18
30 4 16 32
(5) We need to find the difference between the sequence and 2n2.
2n2 d
9 2 -7
12 8 -4
19 18 -1
30 32 +2
(6) The difference here will either be a constant number, in which case the nth term is (1/2a)n2 +d. Or like this case, will itself follow a linear sequence with constant difference, which we should know how to solve.
1 2 3 4
-7, -4, -1, +2
+3 +3 +3
This gives 3n - 10. Therefore the whole formula for the nth term is;
(7) 2n2 + 3n - 10
