C1 January 2011 Q4
4. A sequence \(a_1, a_2, a_3, \ldots\) is defined by\[\begin{aligned}a_1 &= 2\\ a_{n+1} &= 3a_n - c\end{aligned}\]where \(c\) is a constant.
(a) Find an expression for \(a_2\) in terms of \(c\). (1)
Given that \(\displaystyle\sum_{i=1}^{3} a_i = 0\)
(b) find the value of \(c\). (4)
| Scheme | Marks |
|---|---|
| \(\left(a_2 =\right)\ 6 - c\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(a_3 = 3(\text{their } a_2) - c \qquad (= 18 - 4c)\) | M1 |
| \(a_1 + a_2 + a_3 = 2 + \text{"}(6 - c)\text{"} + \text{"}(18 - 4c)\text{"}\) | M1 |
| \(\text{"}26 - 5c\text{"} = 0\) | A1ft |
| So \(c = 5.2\) | A1 o.a.e |
| (4) | |
| (5 marks) |
Notes
1st M1: for attempting \(a_3\). Can follow through their answer to (a) but it must be an expression in \(c\).
2nd M1: for an attempt to find the sum \(a_1 + a_2 + a_3\) must see evidence of sum
1st A1ft: for their sum put equal to 0. Follow through their values but answer must be in the form \(p + qc = 0\)
A1: accept any correct equivalent answer