Prove that the square of an odd number is always 1 more than a multiple of 4. - Edexcel - GCSE Maths - Question 12 - 2018 - Paper 1

To find the square of the odd number, we calculate: $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1.$

From the expression $n^2 = 4(k^2 + k) + 1$, we can see that $4(k^2 + k)$ is a multiple of 4. Therefore, it follows that $n^2$ is always 1 more than this multiple.

Hence, we have proved that the square of an odd number is always 1 more than a multiple of 4.