Answer :

kittymaniac
Call the consecutive integers n, n+1, and n+2. One you find n, you can find the other next two numbers.
n + (n + 1) + (n + 2) = 114
3n + 3 = 114
3n = 111
n = 37

Therefore, the three numbers are n (37), n+1 (38), and n+2 (39).