| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 7 |
| Topic | Geometric Sequences and Series |
| Type | Relationship between two GPs |
| Difficulty | Challenging +1.2 This question requires understanding that squaring terms of a GP creates a new GP with first term a² and ratio r², then manipulating the sum to infinity formula to derive a relationship. Part (b) requires using the constraint r ∈ (-1,1) to find the range of a=2(1+r), which is straightforward substitution. While it involves multiple concepts (GP properties, sum to infinity, algebraic manipulation, and range finding), these are standard techniques without requiring novel insight. The 'show that' format and the range question are typical A-level fare, slightly above average due to the multi-step reasoning across two parts. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<1 |
11.
A geometric sequence, $S _ { 1 }$, has first term $a$ and common ratio $r$ where $a \neq 0$ and $r \in ( - 1,1 )$
A new sequence, $S _ { 2 }$, is formed by squaring each term of $S _ { 1 }$
\begin{enumerate}[label=(\alph*)]
\item Given that the sum to infinity of $S _ { 2 }$ is twice the sum to infinity of $S _ { 1 }$, show that $a = 2 ( 1 + r )$
Fully justify your answer.
\item Determine the set of possible values for $a$.
\section*{Additional Answer Space }
\section*{Additional Answer Space }
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q11 [7]}}