Three consecutive

Let the three consecutive numbers be x+1, x, x-1

Then,
x+1 + x + x - 1 =51

x + x + x + 1 - 1 = 51

3x = 51

x = 17
Also,
 
x - 1 = 17 - 1 = 16
x + 1 = 17 + 1 = 18 

Hence, three consecutive integers are 16, 17 and 18.
Method 2:
Let the three consecutive integers be x, x + 1, x + 2
Sum of these numbers = 51
x + x + 1 + x + 2 = 51
3 x + 3 = 51
3 x = 48
x = 16
Now the numbers are 16, 16 + 1, 16 + 2
Therefore, numbers are 16, 17, 18