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 n^th 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 n^th 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 n^th term is;

(7) 2n² + 3n - 10