| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | October |
| Marks | 5 |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Standard +0.3 This is a straightforward sequences problem requiring basic manipulation of AP and GP formulas. Part (a)(i) is direct substitution, (a)(ii) involves solving a simple quadratic, and (b) follows immediately. The algebraic steps are routine with no conceptual challenges beyond standard A-level content. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
An arithmetic progression has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio $r$.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $r = \frac{a + 2d}{a}$. [1]
\item Find $d$ in terms of $a$. [2]
\end{enumerate}
\item Find the common ratio of the geometric progression. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q12 [5]}}