C1 January 2010 Q7
7. Jill gave money to a charity over a 20-year period, from Year 1 to Year 20 inclusive. She gave £150 in Year 1, £160 in Year 2, £170 in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.
Kevin also gave money to the charity over the same 20-year period.
He gave £\(A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference £30.
The total amount of money that Kevin gave over the 20-year period was twice the total amount of money that Jill gave.
| Scheme | Marks |
|---|---|
| \(a + 9d = 150 + 9\times 10 = 240\) | M1 A1 |
| (2) |
Notes
M: Using \(a + 9d\) with at least one of \(a = 150\) and \(d = 10\).
Being ‘one off’ (e.g. equivalent to \(a + 10d\)), scores M0.
Correct answer with no working scores both marks.
‘Listing’ and other methods
M: Listing terms (found by a correct method with at least one of \(a = 150\) and \(d = 10\)), and picking the 10th term. (There may be numerical slips).
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2}n\left\{2a + (n - 1)d\right\} = \dfrac{20}{2}\left\{2\times 150 + 19\times 10\right\}, \ = 4900\) | M1 A1, A1 |
| (3) |
Notes
M: Attempting to use the correct sum formula to obtain \(S_{20}\), with at least one of \(a = 150\) and \(d = 10\). If the wrong value of \(n\) or \(a\) or \(d\) is used, the M mark is only scored if the correct sum formula has been quoted.
1st A: Any fully correct numerical version.
‘Listing’ and other methods
M: Listing sums, or listing and adding terms (found by a correct method with at least one of \(a = 150\) and \(d = 10\)), far enough to establish the required sum. (There may be numerical slips). Note: 20th term is 340.
A2 (scored as A1 A1) for 4900 (clearly selected as the answer).
If no working (or no legitimate working) is seen, but the answer 4900 is given, allow one mark (scored as M1 A0 A0).
| Scheme | Marks |
|---|---|
| Kevin: \(\dfrac{1}{2}n\left\{2a + (n - 1)d\right\} = \dfrac{20}{2}\left\{2A + 19\times 30\right\}\) | B1 |
| Kevin’s total \(= 2\times\text{"}4900\text{"}\) (or “4900” \(= 2\times\) Kevin’s total) | M1 |
| \(\dfrac{20}{2}\left\{2A + 19\times 30\right\} = 2\times\text{"}4900\text{"}\) | A1ft |
| \(A = 205\) | A1 |
| (4) | |
| (9 marks) |
Notes
B: A correct expression, in terms of \(A\), for Kevin’s total.
M: Equating Kevin’s total to twice Jill’s total, or Jill’s total to twice Kevin’s.
For this M mark, the expression for Kevin’s total need not be correct, but must be a linear function of \(A\) (or \(a\)).
1st A: (Kevin’s total, correct, possibly unsimplified) = 2(Jill’s total), ft Jill’s total from part (b).
By trial and improvement:
Obtaining a value of \(A\) for which Kevin’s total is twice Jill’s total, or Jill’s total is twice Kevin’s (using Jill’s total from (b)): M1
Obtaining a value of \(A\) for which Kevin’s total is twice Jill’s total (using Jill’s total from (b)): A1ft
Fully correct solutions then score the B1 and final A1.
The answer 205 with no working (or no legitimate working) scores no marks.