Higher June 2019 Paper 6 Q13

13 Prove that the mean of any four consecutive even integers is an integer. [4]

Mark scheme

Answer	Marks	Part marks and guidance
\(\dfrac{(2x) + (2x + 2) + (2x + 4) + (2x + 6)}{4}\) \(= \dfrac{8x + 12}{4}\) \(= 2x + 3\) which is an integer OR \((2x) + (2x + 2) + (2x + 4) + (2x + 6)\) \(= 8x + 12\) \(= 4(2x + 3)\) which is divisible by 4 oe	4	M1 for \(2x\), \(2x + 2\), \(2x + 4\) and \(2x + 6\) seen and M1 for adding their four terms in \(x\), e.g. \((2x) + (2x + 2) + (2x + 4) + (2x + 6)\) and M1 for their \((8x + 12) \div 4\) or better, condoning lack of brackets, or for \(4(2x + 3)\) and A1dep (dep on M0M1M1 or M1M1M1) for correct algebraic mean for their four terms and conclusion, e.g. \(2x + 3\) is an integer or \(4(2x + 3)\) which is divisible by 4 If 0 scored, allow SC1 for a numerical example with any 4 consecutive even integers with mean correctly calculated	Or equivalent algebraic representations of 4 consecutive even numbers. In this case, \(x\) does not need to be defined as being an integer. Using \(x\), \(x + 2\), \(x + 4\), \(x + 6\) oe does not score the first M mark unless \(x\) stated as even integer, but can score up to 3 marks for \((x) + (x + 2) + (x + 4) + (x + 6)\), their \((4x + 12) \div 4\) or better, or for \(4(x + 3)\) and the relevant conclusion Using \(x + 1\), \(x + 3\), \(x + 5\), \(x + 7\) oe does not score the first M mark unless \(x\) stated as odd integer but can score up to 3 marks similar to above.