What is the next number in sequence 7 9 12 16?

The first step in working out the equation of a sequence is to write down the differences between the terms. These are:

2, 3, 4...

They are not equal, so we now look at the differences of the differences:

1, 1.

These are equal, so the equation begins x2/2

The sequence - x2/2 is:

6.5, 7, 7.5, 8

So the equation continues x2/2 + x/2

and so by the first term the entire equation is (x2+x+12)/2

This gives us the sequence as:

7, 9, 12, 16, 21, 27, 34...