SOLUTION: the next number in pattern: 3,-6,12,4,20

First thing to do is to always look for a pattern 
in the differences between successive terms:

2nd term - 1st term =  (-6) - (3) = -6-3 =  -9
3rd term - 2nd term = (12) - (-6) = 12+6 = +18
4th term - 3rd term =  (4) - (12) = 4-12 =  -8
5th term - 4th term = (20) -  (4) = 20-4 = +16
 
So the pattern of differences go 

-9, +18, -8, +16

The even term - odd term differences go  -9,  -8,  -7,  -6,  -5, ...
The odd term - even term differences go +18, +16, +14, +12, +10, ...

So 

6th term - 5th term = -7    
      6th term - 20 = -7
           6th term = 13

Edwin