Do not be bothered by the fact that the sum of the two consecutive integers is negative, as the means of answering the problem is identical for both positive and negative answers. Allen above gave the algebraic solution to the problem so I will simply describe a method of thinking about the problem logically and working it out without covering it to an equation.
First any two consecutive integers add up to an odd number. This is by definition because with two consecutive integers one will be positive and one will be negative thus when added together it always produces a negative result.
For example 1,2 give 3 (1+2=3) and 2,3 give 5 (2+3=5).
If you subtract 1 from the sum and divide by 2 you will get the first number in the sequence because as Allen said the equation is n + (n+1) = -105. By subtracting 1 from the sum. In this case -105 - 1 gives = -106. diving by 2 give -53.
Thus the answer is -53 and -52. If the sum had been positive 105 you would subtract 1 and get 104, half of which is 52. Thus the answer would be 52, and 53. As I said it really does make any difference if you are working with positive numbers or negative numbers the answer is the same except for the sign.
-105 yields consecutive integers of -53 and -52 .
105 yields consecutive integers of 52 and 53 .
-1 would yield consecutive integers of -1 and 0 .
1 would yield consecutive integers of 0 and 1 .