C1 June 2007 Q4
4. A girl saves money over a period of 200 weeks. She saves 5p in Week 1, 7p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.
| Scheme | Marks |
|---|---|
| Identify \(a = 5\) and \(d = 2\) (May be implied) | B1 |
| \(\left(u_{200} =\right) a + (200 - 1)d \qquad (= 5 + (200 - 1)\times 2)\) | M1 |
| \(= \underline{403}\)(p) or (£) \(\underline{4.03}\) | A1 |
| (3) |
Notes
B1: can be implied if the correct answer is obtained. If 403 is not obtained then the values of \(a\) and \(d\) must be clearly identified as \(a = 5\) and \(d = 2\).
This mark can be awarded at any point.
M1: for attempt to use \(n\)th term formula with \(n = 200\). Follow through their \(a\) and \(d\).
Must have use of \(n = 200\) and one of \(a\) or \(d\) correct or correct follow through.
Must be 199 not 200.
A1: for 403 or 4.03 (i.e. condone missing £ sign here). Condone £403 here.
N.B.: \(a = 3,\ d = 2\) is B0 and \(a + 200d\) is M0 BUT \(3 + 200\times 2\) is B1M1 and A1 if it leads to 403.
Answer only of 403 (or 4.03) scores 3/3.
ALT Listing
(a) They might score B1 if \(a = 5\) and \(d = 2\) are clearly identified. Then award M1A1 together for 403.
| Scheme | Marks |
|---|---|
| \(\left(S_{200} =\right)\dfrac{200}{2}\left[2a + (200 - 1)d\right]\) or \(\dfrac{200}{2}\left(a + \text{"their 403"}\right)\) | M1 |
| \(= \dfrac{200}{2}\left[2\times 5 + (200 - 1)\times 2\right]\) or \(\dfrac{200}{2}\left(5 + \text{"their 403"}\right)\) | A1 |
| \(= \underline{40\,800}\) or £408 | A1 |
| (3) | |
| (6 marks) |
Notes
M1: for use of correct sum formula with \(n = 200\). Follow through their \(a\) and \(d\) and their 403.
Must have some use of \(n = 200\), and some of \(a\), \(d\) or \(l\) correct or correct follow through.
1st A1: for any correct expression (i.e. must have \(a = 5\) and \(d = 2\)) but can f.t. their 403 still.
2nd A1: for 40800 or £408 (i.e. the £ sign is required before we accept 408 this time).
40800p is fine for A1 but £40800 is A0.
ALT Listing
(b) \(\displaystyle\sum_{r=1}^{200}(2r + 3)\). Give M1 for \(2\times\dfrac{200}{2}\times(201) + 3k\) (with \(k > 1\)), A1 for \(k = 200\) and A1 for 40800.