C1 June 2014 Q5
5. A sequence of numbers \(a_1, a_2, a_3 \ldots\) is defined by\[a_{n+1} = 5a_n - 3, \qquad n \geqslant 1\]Given that \(a_2 = 7\),
| Scheme | Marks |
|---|---|
| \(7 = 5a_1 - 3 \Rightarrow a_1 = ..\) | M1 |
| \(a_1 = 2\) | A1 |
| (2) |
Notes
M1: Writes \(7 = 5a_1 - 3\) and attempts to solve leading to an answer for \(a_1\). If they rearrange wrongly before any substitution this is M0
A1: Cao \(a_1 = 2\)
Special case: Substitutes \(n = 1\) into \(5n - 3\) and obtains answer 2. This is fortuitous and gets M0A0 but full marks are available on (b).
| Scheme | Marks |
|---|---|
| \(a_3 =\) “32” and \(a_4 =\) “157” | M1 |
| \(\displaystyle\sum_{r=1}^{r=4} a_r = a_1 + a_2 + a_3 + a_4\) = “2”+ “7”+ “32”+ “157” | dM1 |
| = 198 | A1 |
| (3) | |
| (5 marks) |
Notes
M1: Attempts to find either their \(a_3\) or their \(a_4\) using \(a_{n+1} = 5a_n - 3,\ a_2 = 7\)
Needs clear attempt to use formula or is implied by correct answers or correct follow through of their \(a_3\)
dM1: (Depends on previous M mark) Sum of their four adjacent terms from the given sequence.
n.b May be given for \(9 + a_3 + a_4\) as they may add 2 + 7 to give 9
(dM0 for sum of an Arithmetic series)
A1: cao 198
Special case
(a) \(a_1 = 32\) is M0 A0
(b) Adds for example 7+32+157 + 782 = or 32+157 + 782 + 3907 is M1 M1 A0
Total mark possible is 2 / 5
(This is not treated as a misread – as it changes the question)