Functional Question Higher (Algebra 5) For the week beginning ….

(a) n is a positive integer. (i) Explain why n(n + 1) must be an even number. (1) (ii) Explain why 2n + 1 must be an odd number. (1) (b) Expand and simplify (2n + 1) 2 Answer (2) (c) Prove that the square of any odd number is always 1 more than a multiple of 8. (3) (Total 7 marks)

(a) (i) Even × odd, so even product or equivalent B1 (ii) 2 × n always even, so 2n + 1 is odd or equivalent B1 (b) 4n 2 + 2n + 2n or 4 terms correct M1 4n 2 + 4n + 1 Must simplify A1 (c) Odd 2 – 1 = (2n + 1) 2 –1 = 4n 2 + 4n = 4n(n + 1) Must factorise B1 = 4 × even Deduce ‘even’ connection B1 = multiple of 8 Concluding statement B1 [7]

In part (a) some candidates were confused by positive/negative and odd/even and could not offer any sensible reasoning. Whilst many good answers were seen (usually more for (ii) than (i)) there were also too many ‘partial’ solutions ie. only considering what happened when n was even and not continuing the argument for when n was odd. Such incomplete answers did not score the mark(s). Part (b) was well done by the vast majority of candidates; some made a mistake in one of the terms gaining just 1 of the 2 marks. Part (c) was meant to be a test of candidates’ ability to handle a ‘using and applying’ approach. The question did try to lead them into the algebraic approach that was necessary to gain any marks at all but even the most able candidates did not take the hint. There were some correct solutions but they probably number far less than 1% of the total entry. It is important that future candidates are shown the difference between ‘verify’ and ‘prove’. Common Mistakes – what did the examiners say?