Question

prove that the difference between the squares of two consecutive natural number is equal to their sum.

Answer 1

Let the two consecutive natural numbers be x and x+1

Difference between the squares of two consecutive natural numbers is:-
(x)²-(x+1)²
=x²-[x²+1¹+2(x)(1)]
=x²-(x²+1+2x)
=x²-x²+1+2x
=2x+1

The sum of consecutive natural numbers is;-
x+(x+1)
=x+x+1
=2x+1

Both are equal. Hence proved

For example,
Let the two consecutive natural numbers be 2 and 3
Difference between the squares of those numbers is:-
(3)²-(2)²
=9-4
=5
Sum of those consecutive natural numbers is:-
2+3
=5

So proved.

Hope it helps