| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 This is a straightforward geometric series question requiring knowledge of convergence conditions (|r| < 1) and the sum to infinity formula. Part (a) involves solving an absolute value inequality, and part (b) uses S∞ = a/(1-r) to find x. Both are standard textbook exercises with clear methods and no novel insight required, making it slightly easier than average. |
| Spec | 1.02h Express solutions: using 'and', 'or', set and interval notation1.04i 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.
It is given that the geometric series
$$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$
is convergent.
\begin{enumerate}[label=(\alph*)]
\item Find the set of possible values of $x$, giving your answer in set notation. [5]
\item Given that the sum to infinity of the series is $\frac{2}{3}$, find the value of $x$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2024 Q8 [8]}}