| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | October |
| Marks | 6 |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.8 Part (a) involves straightforward recursive substitution requiring only basic arithmetic (squaring and subtracting 5), then observing the sequence diverges. Part (b) is a standard GP problem: find the common ratio from consecutive terms (r=2/3), find the first term (a=18), then apply the sum to infinity formula. Both parts are routine textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
\begin{enumerate}[label=(\alph*)]
\item A sequence has terms $u_1, u_2, u_3, \ldots$ defined by $u_1 = 3$ and $u_{n+1} = u_n^2 - 5$ for $n \geq 1$.
\begin{enumerate}[label=(\roman*)]
\item Find the values of $u_2$, $u_3$ and $u_4$. [2]
\item Describe the behaviour of the sequence. [1]
\end{enumerate}
\item The second, third and fourth terms of a geometric progression are 12, 8 and $\frac{16}{3}$.
Determine the sum to infinity of this geometric progression. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q4 [6]}}