Prove algebraically that 
(2n + 1)^2 – (2n + 1) is an even number for all positive integer values of n. (3 marks)

We can show something is even if it is a multiple of two, as every multiple of two is even. It is useful to see certain tricks, and I will aim to teach you these in my tutorials, these tricks make problems easier and will save you time in your lessons and exams! (2n + 1)² – (2n + 1) = (2n + 1) [(2n + 1) – 1] = (2n + 1) [2n] = 2 n(2n+1).As (2n + 1)² – (2n + 1) is a multiple of two (as it is equal to 2n (2n+1)), we have shown that it is an even number for all positive integer values of n.