Question

Prove that the difference between the squares of any two consecutive integers is equal to the sum of these integers.

Answer 1

The difference between the square of any two integers is equal to the sum of the integers.

Let x be the first integer and the next integer will be x+ 1.

The difference in the square of the integer will be

(x+1)² - x²

=( x+1)(x+1) - x²

= (x² + x + x + 1) - x²

= x² + 2x + 1 - x²

= 2x + 1

The addition of the integers will be

x + x + 1 = 2x + 1

Therefore;

2x + 1 = 2x + 1

Therefore, The difference between the square of any two integers is equal to the sum of the integers.