Prove that the difference of the square of two consecutive odd numbers is always a multiple of 8. [OCR GCSE June 2017 Paper 5]

Part 1 of this question asks you to explain why 2n+1 is an odd number, so it is assumed that the student knows this already. The definition of any odd number is 2n+1. Since all consecutive odd numbers are two values apart, the next consecutive odd number is defined as 2n+3 (for all n). The square of the equations are: (2n+1)²=4n²+4n+1 (2n+3)²=4n²+12n+9. Then to find the difference we must subtract one equation from the other. It doesn't matter which way round you do this, the result will be ±(8n-8) = ±8(n-1). This solution shows that no matter what n is, it is being multiplied by 8: the result (the difference of the square of two consecutive odd numbers) is therefore always a multiple of 8.