3,   -6,   12,    4,   20,... 

The odd-numbered terms go:

3,         12,         20,                   

3+9=12
12+8=20
20+7=27, so continuing the pattern of adding 1 less each time:
27+6=33
33+5=38

3,         12,         20,        27,         33,         38,... 

 

The even terms go

  -6,     4,

-6+10 = 4
 4+9 = 13  continuing the pattern of adding 1 less each time.
13+8 = 21
21+7 = 28
28+6 = 34

So the even-numbered terms go:

     -6,          4,        13,         21,         34,      ...

Put the odd-numbered terms and the even-numbered 
terms together:

3,   -6,   12,    4,   20,  13,   27,   21,   33,   34,   38,...

That's a pattern.  It may not be the pattern your teacher had in 
mind, but it's just as good as his.   There are many possible 
patterns for any sequence.  It beats me how they can expect
you to come up the same pattern that they have in mind. 

Edwin