| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 This is a straightforward application of geometric series convergence conditions (|r| < 1) followed by using the sum to infinity formula. Part (a) requires solving an absolute value inequality, and part (b) uses S∞ = a/(1-r). Both are standard textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
8. In this question you must show detailed reasoning.
It is given that the geometric series
$$1 + \frac { 5 } { 3 x - 4 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 2 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 3 } + \ldots$$
is convergent.
\begin{enumerate}[label=(\alph*)]
\item Find the set of possible values of $x$, giving your answer in set notation.
\item Given that the sum to infinity of the series is $\frac { 2 } { 3 }$, find the value of $x$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q8 [8]}}