Question 631039
If the first number is x then the next natural number will be x + 1.
The sum of the squares will therefore be
x^2 + (x + 1)^2
x^2 + x^2 + 2x + 1
ie 2x^2 + 2x + 1
This must equal 113 so
2x^2 + 2x + 1 = 113
Subtract 113 from each side of the equation to get
2x^2 + 2x + 1 -113 = 113 - 113
2x^2 + 2x - 112 = 0
Take out a common factor of 2 and divide each side of the equation by 2 to get
x^2 + x - 56 = 0
This factorises to get 
(x + 8)(x - 7)= 0
ie x = -8 or 7
The numbers must be natural numbers so that excludes -8.
ie x must be 7 and the other consecutive number must be 8.