Higher November 2021 Paper 2 Q10
10.
(a) Find the next term of each of the following sequences.
(i) 2, 9, 11, 20, 31, 51, … [1]
(ii) \(1,\ \ \sqrt{2},\ \ 2,\ \ 2\sqrt{2},\ \ 4,\ \ 4\sqrt{2},\ \ 8,\ \ \ldots\) [1]
(b) Find the \(n\)th term of the following sequence.
2, 6, 12, 20, 30, … [2]
2, 6, 12, 20, 30, … [2]
(c) The numbers 4, 6, and 8 are three consecutive even integers.
Complete the following proof to show that the sum of any three consecutive even integers is a multiple of 6. [3]
Proof:
Every even number is a multiple of 2.
Let the smallest of the three even numbers be \(2n\), where \(n\) is an integer.
The second of the three even numbers must be ……………………………………
and the third of the three even numbers must be ……………………………………
Therefore ……………………………………
Complete the following proof to show that the sum of any three consecutive even integers is a multiple of 6. [3]
Proof:
Every even number is a multiple of 2.
Let the smallest of the three even numbers be \(2n\), where \(n\) is an integer.
The second of the three even numbers must be ……………………………………
and the third of the three even numbers must be ……………………………………
Therefore ……………………………………
| Scheme | Marks | Comments |
|---|---|---|
| (i) 82 | B1 | |
| (ii) \(8\sqrt{2}\) | B1 |
| Scheme | Marks | Comments |
|---|---|---|
| \(n^2 + n\) oe | B2 | Accept unsimplified e.g. \(n \times n + n\) B1 for \(n^2 + kn + c\) where \(k\) and \(c\) are not both 0 |
| Scheme | Marks | Comments |
|---|---|---|
| \(2n + 2\) and \(2n + 4\) | B1 | oe |
| \((2n + 2n + 2 + 2n + 4 =)\ 6n + 6\) | B1 | FT \(2n\) + ‘their \(2n + 2\)’ + ‘their \(2n + 4\)’ |
| Convincing statement of divisibility by 6 e.g. \(= 6(n + 1)\) or ‘both terms divide by 6’ | E1 | oe |
| (7) |