| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | October |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Find N for S_∞ - S_N condition |
| Difficulty | Standard +0.8 This question requires understanding of geometric series (sum to infinity formula), algebraic manipulation of the inequality involving the tail sum, and solving a logarithmic inequality to find N. While the geometric series is standard A-level content, the problem requires multiple conceptual steps: recognizing the tail sum structure, applying the sum to infinity formula correctly, rearranging the inequality, and using logarithms with careful attention to inequality direction. The 8-mark allocation and requirement for 'detailed reasoning' confirms this is above-average difficulty, though not exceptionally hard for a capable student. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
In this question you must show detailed reasoning.
A sequence $t_1, t_2, t_3 \ldots$ is defined by $t_n = 25 \times 0.6^n$.
Use an algebraic method to find the smallest value of $N$ such that
$$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]
\hfill \mbox{\textit{SPS SPS SM 2020 Q10 [8]}}