Consecutive Whole Numbers: x + y = y2 - x2

[Dennis Plews]

The following popped into my head and I am curious whether it is already a known relationship and whether it has an utility in math/physics. It is a follows: Where x and y are consecutive, whole numbers, the following is true: x + y = y² - x²

 

[TeethWhitener]

If x and y are consecutive, then y=x+1.
x+y=x+x+1=2x+1
y²-x²=(x+1)²-x²=x²+2x+1-x²=2x+1
The proof is trivial. I can't think of any immediate application in math or physics.
(For a slightly less trivial, but much intuitively prettier proof, think about this geometrically as a difference of the areas of two squares with sides x and x+1.)

 

[mfb]

Or, even shorter: y²-x² = (y-x)(y+x)
By definition y-x=1, so y²-x² = (y-x)(y+x) = y+x = x+y.

Nothing new.