Break into two sub-sequences taking every other term:

  5,10, 3,6,-1,-2,-9      <--original sequence

  5,    3,  -1,   -9      <--first sub-sequence

    10,   6,   -2,        <--second sub-sequence

First sequence:    5,3,-1,-9,...

            5      <--start with 5 
  5 -  2 =  3
  3 -  4 = -1
 -1 -  8 = -9       so we continue the pattern of doubling what we subtract
 -9 - 16 = -25
-25 - 32 = -57

Second sequence:  10,6,-2,... 

           10      <-- start with 10
 10 -  4 =  6     
  6 -  8 = -2       so we continue the pattern of doubling what we subtract
 -2 - 16 = -18
-18 - 32 = -50
-50 - 64 = -114

  
  5,    3,  -1,   -9,    -25,    -57        <--first sub-sequence

    10,   6,   -2,   -18,    -50     -114   <--second sub-sequence 

  5,10, 3,6,-1,-2,-9,-18,-25,-50,-57,-114   <---original sequence 

That's 5 more terms.  You only wanted 3.

Edwin