Higher November 2019 Paper 3 Q15
15 Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4 (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| Proof | M1 | for correct expressions for two consecutive even numbers eg \(2n\) and \(2n + 2\) |
| M1 | (dep M1) for expanding both expressions with at least one expansion fully correct eg \(4n^2\) and \(4n^2 + 4n + 4n + 4\) or for factorising both terms and intention to square correctly eg \((2n)^2\) and \(2^2(n + 1)^2\) | |
| A1 | complete proof |
Additional guidance
\((2n)^2 + (2n + 2)^2\)
\(= 4n^2 + 4n^2 + 8n + 4\)
\(= 8n^2 + 8n + 4 = 4(2n^2 + 2n + 1)\)
Or
\((2n)^2 + (2n - 2)^2\)
\(= 4n^2 + 4n^2 - 8n + 4\)
\(= 8n^2 - 8n + 4 = 4(2n^2 - 2n + 1)\)
Or
\((2n)^2 + (2n + 2)^2\)
\(= 4(n)^2 + 4(n + 1)^2\)
\(= 4(n^2 + (n + 1)^2)\)