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

The square of this odd number can be calculated as follows:

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

From the squared expression, we can rewrite it as:

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

Here, $4(k^2 + k)$ is a multiple of 4.

Thus, we observe that:

$n^2 = 4m + 1$

where $m = k^2 + k$, which is an integer. Hence, we conclude that the square of an odd number is always 1 more than a multiple of 4.