| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | October |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Find N for S_∞ - S_N condition |
| Difficulty | Challenging +1.8 This question requires recognizing that the sequence t_n = 5-2n creates exponentially decreasing terms in 2^{t_n}, converting to a geometric series, finding the sum to infinity formula, then solving an inequality involving logarithms to find N. It combines multiple A-level topics (sequences, geometric series, logarithms, inequalities) in a non-routine way that requires genuine problem-solving rather than pattern-matching to standard exercises. The 8-mark allocation and 'show detailed reasoning' instruction confirm this is a substantial multi-step problem, though it remains within reach of strong A-level students. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04i 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 $t_n = 5 - 2n$.
Use an algebraic method to find the smallest value of $N$ such that
$$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]
\hfill \mbox{\textit{SPS SPS FM 2020 Q9 [8]}}