Prove that the square of an odd number is always 1 more than a multiple of 4. [Edexcel Higher Tier 2018, Paper 1: Question 12]

We want to start with “the square of an odd number” and show something. Remember that any odd number can be expressed as 2n+1 (an even number is 2n, and any odd number is 1 larger than an even). So squaring an odd number gives us (2n+1)².Let’s see if we can evaluate this algebraically to get what the question asks for. Firstly we need to expand the brackets:(2n+1)² = (2n)² + 2(2n)(1) + 1²            [by hairpin/FOIL method or the trick for squaring brackets] = 4n² + 4n + 1Now notice the factor of 4 in two of the terms; we can factorise! = 4(n²+n) + 1which is something times 4, plus 1; i.e. “1 more than a multiple of 4”. This is the proof the question required, and is general for any odd number 2n+1.