Higher November 2018 Paper 3 Q15
15 Prove algebraically that the difference between the squares of any two consecutive odd numbers is always a multiple of 8 (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| proof | C1 | for writing an expression for an odd number, eg \(2n + 1\) or \(2n - 1\) (assuming \(n\) is any integer) or states \(n\) is even and eg \((n + 1)\) or \((n + 3)\) as odd numbers |
| C1 | for a correct expression of the form \((2n + 1)^2 - (2n - 1)^2\) expanded eg \(4n^2 + 12n + 9 - (4n^2 + 4n + 1)\) or \(4n^2 + 4n + 1 - (4n^2 - 4n + 1)\) or \((2n + 1 + 2n - 1)(2n + 1 - (2n - 1))\) or when \(n\) is even and eg \((n^2 + 6n + 9) - (n^2 + 2n + 1)\) (\(= 4n + 8\)) | |
| C1 | for a correct simplified expression as a multiple of 8 eg \(8n + 8\) or \(8n\) or when \(n\) is even and eg \(4n + 8\) and full explanation as to why \(4(n + 2)\) is always a multiple of 8 |
Additional guidance
Expansion of \((2n - 1)^2 - (2n + 1)^2\) oe is acceptable