A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers - AQA - A-Level Maths Mechanics - Question 7 - 2017 - Paper 2

To prove that the sum of two consecutive odd numbers can be expressed as the difference of two square numbers, we can represent two consecutive odd numbers algebraically.

Let the first consecutive odd number be represented as:

• The first odd number can be written as: $2n + 1$
• The second odd number can be written as: $2n + 3$

Their sum is:

$(2n + 1) + (2n + 3) = 4n + 4$

Next, we express this sum as the difference of two squares:

$n^2 + 2n + 1 = (n + 1)^2$
$n^2 = n^2$

Thus we can write:

$(n + 1)^2 - n^2 = 4n + 4$

This shows that the sum of two consecutive odd numbers can indeed be expressed as the difference between two square numbers.