| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Simple recurrence evaluation |
| Difficulty | Moderate -0.3 This is a straightforward recurrence relation with complex numbers requiring repeated multiplication by i, pattern recognition of a cycle of period 4, and summation using that periodicity. While it involves complex numbers, the mechanics are routine—multiply by i repeatedly, spot the cycle, then use it to sum. Slightly easier than average due to the simple recurrence and clear pattern. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.04e Line intersections: parallel, skew, or intersecting |
**Question 4: Complex sequence $u_1 = 1+\mathrm{i}$, $u_{n+1} = \mathrm{i}u_n$**
**(i)**
- $u_2 = \mathrm{i}(1+\mathrm{i})$, $u_3 = \mathrm{i}(-1+\mathrm{i})$ or $\mathrm{i}(\mathrm{i} + \mathrm{i}^2)$ oe **M1** Attempt correct process to find at least $u_2$ and $u_3$
- $u_2 = -1 + \mathrm{i},\ u_3 = -1 - \mathrm{i}$ **A1** Correct, simplified, $u_2$ and $u_3$
- $u_4 = 1 - \mathrm{i},\ u_5 = 1 + \mathrm{i},\ u_6 = -1 + \mathrm{i}$ **A1** Fully correct and simplified
**(ii)**
- Periodic (with period 4) **B1** Any equivalent description. Allow geometric.
**Total: 4 marks**4 A sequence of complex numbers is defined by
$$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$
(i) Find $u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }$ and $u _ { 6 }$.\\
(ii) Describe the behaviour of the sequence.\\
(iii) Hence evaluate $\sum _ { n = 1 } ^ { 73 } u _ { n }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2017 Q4 [4]}}