Standard +0.8 This question requires understanding of geometric series, finding the sum to infinity formula, setting up an inequality involving the tail of the series, and solving a logarithmic inequality to find N. While the geometric series itself is standard Further Maths content, the requirement to find when the tail sum falls below a threshold requires careful algebraic manipulation and logarithm work, making it moderately challenging but still within typical FM scope.
In this question you must show detailed reasoning.
A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]
In this question you must show detailed reasoning.
A sequence $u_1, u_2, u_3 \ldots$ is defined by $u_n = 25 \times 0.6^n$.
Use an algebraic method to find the smallest value of $N$ such that $\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}$. [7]
\hfill \mbox{\textit{SPS SPS FM 2024 Q7 [7]}}