4MA0 Higher June 2015 Paper 4 Q20
20 \(n\) is a positive integer.
(a) Explain why \(2n + 1\) is an odd number for all values of \(n\). (1)
(b) Show, using algebra, that the sum of any 4 consecutive odd numbers is always a multiple of 8 (3)
| Scheme | Marks |
|---|---|
| Eg. \(2n\) is always even so \(2n + 1\) is odd Answer: explanation | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \((2n + 1) + (2n + 3) + (2n + 5) + (2n + 7)\) oe | M1 |
| \(8n + 16\) | M1dep |
| \(8(n + 2)\) Answer: show | A1 |
| (3) |
Notes
A1: or a complete explanation from correct algebraic working
Alternative
| Scheme | Marks |
|---|---|
| let \(x\) be an even number \(x + 1 + x + 3 + x + 5 + x + 7\) oe | M1 |
| \(4x + 16\) | M1dep |
| \(4(x + 4)\) | A1 |
Notes
M1: For defining \(x\) (at beginning or end) and summing 4 consecutive odd numbers
A1: For a complete explanation from correct algebraic working., eg \(x + 4\) must be even and 4 times an even number = 4 × 2\(n\) which is a multiple of 8
Alternative
| Scheme | Marks |
|---|---|
| Let \(y\) be an odd number \(y + y + 2 + y + 4 + y + 6\) oe | M1 |
| \(4y + 12\) | M1dep |
| \(4(y + 3)\) | A1 |
Notes
M1: For defining \(y\) (at beginning or end) and summing 4 consecutive odd numbers
A1: For a complete explanation from correct algebraic working. eg \((y + 3)\) is (odd + odd) which is even and therefore a multiple of 2, and 4 × 2n is a multiple of 8