| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.8 Part (a) is a standard textbook proof of the geometric series formula that appears in every A-level syllabus, requiring only algebraic manipulation of S_n - rS_n. Part (b) involves substituting the formula and solving a simple equation (4S_5 = S_10), leading to a quadratic. While it requires careful algebra, this is a routine exercise with no novel problem-solving required, making it easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum |
9. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
A geometric series has common ratio $r$ and first term $a$.\\
Given $r \neq 1$ and $a \neq 0$
\begin{enumerate}[label=(\alph*)]
\item prove that
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
Given also that $S _ { 10 }$ is four times $S _ { 5 }$
\item find the exact value of $r$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q9 [8]}}