C1 June 2008 Q7
7. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km.
On the \(n\)th Saturday Sue runs 43 km.
| Scheme | Marks |
|---|---|
| 5, 7, 9, 11 or 5+2+2+2=11 or 5+6=11 use \(a = 5\), \(d = 2\), \(n = 4\) and \(t_4 = 5 + 3\times 2 = 11\) | B1 |
| (1) |
Notes
B1: Any other sum must have a convincing argument
| Scheme | Marks |
|---|---|
| \(t_n = a + (n - 1)d\) with one of \(a = 5\) or \(d = 2\) correct (can have a letter for the other) | M1 |
| \(= 5 + 2(n - 1)\) or \(2n + 3\) or \(1 + 2(n + 1)\) | A1 |
| (2) |
Notes
M1: for an attempt to use \(a + (n - 1)d\) with one of \(a\) or \(d\) correct (the other can be a letter)
Allow any answer of the form \(2n + p\ (p \neq 5)\) to score M1.
A1: for a correct expression (needn’t be simplified) [ Beware \(5 + (2n - 1)\) scores A0]
Expression must be in \(n\) not \(x\).
Correct answers with no working scores 2/2.
Do not give credit for part (b) if the equivalent work is given in part (d)
Poor labelling may occur (especially in (b) and (c) ). If you see work to get \(n(n + 4)\) mark as (c)
| Scheme | Marks |
|---|---|
| \(S_n = \dfrac{n}{2}\left[2\times 5 + 2(n - 1)\right]\) or use of \(\dfrac{n}{2}\left(5 + \text{"their } 2n + 3\text{"}\right)\) (may also be scored in (b)) | M1A1 |
| \(= \left\{n(5 + n - 1)\right\} = n(n + 4) \quad (*)\) | A1cso |
| (3) |
Notes
M1: for an attempt to use \(S_n\) formula with \(a = 5\) or \(d = 2\) or \(a = 5\) and their “\(2n + 3\)”
1st A1: for a fully correct expression
2nd A1: for correctly simplifying to given answer. No incorrect working seen. Must see \(S_n\) used.
Poor labelling may occur (especially in (b) and (c) ). If you see work to get \(n(n + 4)\) mark as (c)
| Scheme | Marks |
|---|---|
| \(43 = 2n + 3\) | M1 |
| \([n] = 20\) | A1 |
| (2) |
Notes
M1: for forming a suitable equation in \(n\) (ft their (b)) and attempting to solve leading to \(n = \ldots\)
A1: for 20
Correct answer only scores 2/2. Allow 20 following a restart but check working.
eg \(43 = 2n + 5\) that leads to \(40 = 2n\) and \(n = 20\) should score M1A0.
NB “attempting to solve” eg part (d) means we will allow sign slips and slips in arithmetic but not in processes. So dividing when they should subtract etc would lead to M0.
Listing in parts (d) and (e) can score 2 (if correct) or 0 otherwise in each part.
| Scheme | Marks |
|---|---|
| \(S_{20} = 20\times 24,\ = \underline{480}\) (km) | M1A1 |
| (2) | |
| (10 marks) |
Notes
M1: for using their answer for \(n\) in \(n(n + 4)\) or \(S_n\) formula, their \(n\) must be a value.
A1: for 480 (ignore units but accept 480 000 m etc)[ no matter where their 20 comes from]
Listing in parts (d) and (e) can score 2 (if correct) or 0 otherwise in each part.