June 2024 Paper 1 Q12
12 The terms, \(u_n\), of a periodic sequence are defined by
\[u_1 = 3 \quad \text{and} \quad u_{n+1} = \frac{-6}{u_n}\](a) Find \(u_2\), \(u_3\) and \(u_4\) [2 marks]
(b) State the period of the sequence. [1 mark]
(c) Find the value of \(\displaystyle\sum_{n=1}^{101} u_n\) [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(u_1 = 3\) into \(\dfrac{-6}{u_n}\) PI \(u_2 = -2\) | M1 | 1.1a |
| Obtains \(u_2 = -2, u_3 = 3, u_4 = -2\) Condone missing labels if order is obvious. | A1 | 1.1b |
| (2) |
Typical solution
\[u_2 = -2\]\[u_3 = 3\]\[u_4 = -2\]| Scheme | Marks | AO |
|---|---|---|
| States 2 | B1 | 2.2a |
| (1) |
Typical solution
2
| Scheme | Marks | AO |
|---|---|---|
| Shows that pairs of consecutive terms sum to 1 in a series Or Considers a sum of 3s and a sum of \(\pm 2\)s | M1 | 3.1a |
| Deduces \(\displaystyle\sum_{n=1}^{101} u_n = 53\) | R1 | 2.2a |
| (2) | ||
| (5 marks) |