Proof

Sample questions

Original practice questions written in the style of the exam.

1. Question 14 marks

   Prove that the sum of the squares of any two consecutive odd integers is always 2 more than a multiple of 8.

   Mark scheme

   • B1 Writes two consecutive odd integers as $2n + 1$ and $2n + 3$ for integer $n$ (1)
   • M1 Expands $(2n + 1)^2 + (2n + 3)^2$ (1)
   • A1 $= 8n^2 + 16n + 10$ (1)
   • A1 $= 8(n^2 + 2n + 1) + 2$, and $n^2 + 2n + 1$ is an integer, so the sum is 2 more than a multiple of 8 (1)