SOLUTION: what is the next number in the following sequence 3,-6,12,4,20

2nd term - 1st term = (-6)-(3) = -9
3rd term - 2nd term = (12)-(-6) = 12+6 = 18
4th term - 3rd term = (4)-(12) = -8
5th term - 4th term = (20)-(4) = 16

It looks like the odd-numbered differences 
are going 

-9, -8, -7, -6, -5, ...

And it looks like the even numbered 
differences are going 

18, 16, 14, 12, 10...

If so then let the 6th term be x

2nd term - 1st term = (-6)-(3) = -9
3rd term - 2nd term = (12)-(-6) = 12+6 = 18
4th term - 3rd term = (4)-(12) = -8
5th term - 4th term = (20)-(4) = 16
6th term - 5th term = (x)-(16) = -7

(x)-(16) = -7
    x-16 = -7
       x = 16-7
       x = 9

So the 6th term is 9

Just for fun, let's get one more term. 
Let the 7th term be y.

2nd term - 1st term = (-6)-(3) = -9
3rd term - 2nd term = (12)-(-6) = 12+6 = 18
4th term - 3rd term = (4)-(12) = -8
5th term - 4th term = (20)-(4) = 16
6th term - 5th term = (9)-(16) = -7
7th term - 6th term = (y)-(-7) = 14

(y)-(-7) = 14
     y-7 = 14
       y = 21

So the sequence goes:

3,-6,12,4,20,9,21

Edwin