| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2026 |
| Session | November |
| Marks | 8 |
| Topic | Geometric Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.3 Part (a) is a standard proof of the geometric series formula that appears in most A-level textbooks and requires only algebraic manipulation of Sₙ - rSₙ. Part (b) requires substituting the formula and solving 4S₅ = S₁₀, leading to a factorizable equation. While it involves multiple steps and algebraic manipulation, both parts are routine Further Maths content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum |
In this question you must show all stages of your working.
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
$$S_n = \frac{a(1-r^n)}{1-r}$$
[4]
\end{enumerate}
Given also that $S_{10}$ is four times $S_5$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the exact value of $r$.
[4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2026 Q9 [8]}}