| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 3 |
| Topic | Geometric Sequences and Series |
| Type | GP with trigonometric terms |
| Difficulty | Challenging +1.2 This question requires recognizing that cos(180n)° = (-1)^n, converting to a geometric series sum from n=2 to infinity with first term (1/4)²·1 = 1/16 and common ratio -1/4, then applying the standard formula. While it involves infinite series and a trigonometric simplification, it's a fairly direct application of A-level techniques with minimal problem-solving insight required. The 3-mark allocation confirms it's a routine exercise, though slightly above average difficulty due to the infinite series context. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<11.05a Sine, cosine, tangent: definitions for all arguments |
Show that
$$\sum_{n=2}^{\infty} \left(\frac{1}{4}\right)^n \cos(180n)^{\circ} = \frac{9}{28}$$
[3]
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q8 [3]}}