C1 June 2014 (R) Q3
3. A sequence \(a_1, a_2, a_3, \ldots\) is defined by\[\begin{aligned} a_{n+1} &= 4a_n - 3, \qquad && n \geqslant 1 \\ a_1 &= k, && \text{where } k \text{ is a positive integer.} \end{aligned}\]
(a) Write down an expression for \(a_2\) in terms of \(k\). (1)
Given that \(\displaystyle\sum_{r=1}^{3} a_r = 66\)
(b) find the value of \(k\). (4)
| Scheme | Marks |
|---|---|
| \((a_2 =)\ \ 4k - 3\) | B1 |
| (1) |
Notes
B1: \(4k - 3\) cao
| Scheme | Marks |
|---|---|
| \(a_3 = 4(4k - 3) - 3\) | M1 |
| \(\displaystyle\sum_{r=1}^{3} a_r = k + 4k - 3 + 4(4k - 3) - 3 = ..k \pm \ldots\) | M1 |
| \(21k - 18 = 66 \Rightarrow k = \ldots\) | dM1 |
| \(k = 4\) | A1 |
| (4) | |
| (5 marks) |
Notes
M1: An attempt to find \(a_3\) from iterative formula \(a_3 = 4a_2 - 3\). Condone bracketing errors for the M mark
M1: Attempt to sum their \(a_1, a_2\) and \(a_3\) to get a linear expression in \(k\) (Sum of Arithmetic series is M0)
dM1: Sets their linear expression to 66 and solves to find a value for \(k\). It is dependent upon the previous M mark
A1: cao \(k = 4\)