C1 June 2009 Q7
7. A sequence \(a_1, a_2, a_3, \ldots\) is defined by\[\begin{aligned} a_1 &= k, \\ a_{n+1} &= 2a_n - 7, \qquad n \geqslant 1, \end{aligned}\]where \(k\) is a constant.
Given that \(\displaystyle\sum_{r=1}^{4} a_r = 43\),
| Scheme | Marks |
|---|---|
| \((a_2 =)\,2k - 7\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \((a_3 =)\,2(2k - 7) - 7\) or \(4k - 14 - 7\), \(= 4k - 21 \qquad\) (*) | M1, A1cso |
| (2) |
Notes
M1: must see 2(their \(a_2\)) \(- 7\) or \(2(2k - 7) - 7\) or \(4k - 14 - 7\). Their \(a_2\) must be a function of \(k\).
A1cso: must see the \(2(2k - 7) - 7\) or \(4k - 14 - 7\) expression and the \(4k - 21\) with no incorrect working
| Scheme | Marks |
|---|---|
| \((a_4 =)\,2(4k - 21) - 7 \quad (= 8k - 49)\) | M1 |
| \(\displaystyle\sum_{r=1}^{4} a_r = k + \text{"}(2k - 7)\text{"} + (4k - 21) + \text{"}(8k - 49)\text{"}\) | M1 |
| \(k + (2k - 7) + (4k - 21) + (8k - 49) = 15k - 77 = 43 \qquad k = 8\) | M1 A1 |
| (4) | |
| (7 marks) |
Notes
1st M1: for an attempt to find \(a_4\) using the given rule. Can be awarded for \(8k - 49\) seen.
Use of formulae for the sum of an arithmetic series scores M0M0A0 for the next 3 marks.
2nd M1: for attempting the sum of the 1st 4 terms. Must have “+” not just , or clear attempt to sum.
Follow through their \(a_2\) and \(a_4\) provided they are linear functions of \(k\).
Must lead to linear expression in \(k\). Condone use of their linear \(a_3 \ne 4k - 21\) here too.
3rd M1: for forming a linear equation in \(k\) using their sum and the 43 and attempt to solve for \(k\) as far as \(pk = q\)
A1: for \(k = 8\) only so \(k = \dfrac{120}{15}\) is A0
Answer Only (e.g. trial improvement)
Accept \(k = 8\) only if \(8 + 9 + 11 + 15 = 43\) is seen as well
Sum \(a_2 + a_3 + a_4 + a_5\) or \(a_2 + a_3 + a_4\)
Allow: M1 if \(8k - 49\) is seen, M0 for the sum (since they are not adding the 1st 4 terms) then M1 if they use their sum along with the 43 to form a linear equation and attempt to solve but A0