How do you find the nth term formula for a sequence with non-constant difference?

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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

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