Question

The sum of two consecutive numbers is S. The square of the larger number minus the square of the smaller number is?

Answer 1

let the two numbers be x and x+1

x + x+1 = S
2x = S - 1
x = (S-1)/2

The square of the larger number minus the square of the smaller number
= (x+1)^2 - x^2
= x^2 + 2x + 1 - x^2
= 2x + 1
= (S-1) + 1
= S

check:
let's take 37, and 38
sum = 75
square of larger =1444
square of smaller = 1369
1444 - 1369 = 75 , which is the sum of the 2 numbers.

Answer 2

This can also be seen by noting that

a^2-b^2 = (a+b)(a-b)

In this case, since the numbers are consecutive, that is just

S*1 = S