Higher January 2022 Paper 1R Q15
15 Using algebra, prove that, given any 3 consecutive whole numbers, the sum of the square of the smallest number and the square of the largest number is always 2 more than twice the square of the middle number.
(3)
| Scheme | Marks |
|---|---|
| E.g. \(n\), \(n + 1\), \(n + 2\) \((n^2 =)\ n^2\) \(((n + 1)^2 =)\ n^2 + n + n + 1 = n^2 + 2n + 1\) oe \(((n + 2)^2 =)\ n^2 + 2n + 2n + 4 = n^2 + 4n + 4\) oe or E.g. \(n - 1\), \(n\), \(n + 1\) \(((n - 1)^2 =)\ n^2 - n - n + 1 = n^2 - 2n + 1\) oe \((n^2 =)\ n^2\) \(((n + 1)^2 =)\ n^2 + n + n + 1 = n^2 + 2n + 1\) oe | M1 |
| \(n^2 + n^2 + 2n + 2n + 4\ (= 2n^2 + 4n + 4)\) oe and \(2(n + 1)^2 = 2n^2 + 2n + 2n + 2\ (= 2n^2 + 4n + 2)\) oe or \(n^2 - 2n + 1 + n^2 + 2n + 1\ (= 2n^2 + 2)\) oe | M1 |
| E.g. \(2n^2 + 4n + 4 = 2n^2 + 4n + 2 + 2\) oe or \(2(x + 1)^2 + 2 = 2(x + 1)^2 + 2\) oe or \(2n^2 + 2 = 2n^2 + 2\) oe Answer: Complete proof | A1 |
| (3) | |
| (3 marks) |
Notes
M1: for 3 appropriate terms for their 3 numbers and for correctly finding the expansion of at least 2 squares
(Allow 2 × middle number + 2)
M1: for finding the sum of first and last square and double the square of the middle
(Allow 2 × middle number + 2)
A1: for conclusion from two correct expressions
e.g. \(2n^2 + 4n + 4\) and \(2n^2 + 4n + 2\)