| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | November |
| Marks | 8 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Recurrence relation solving for closed form |
| Difficulty | Standard +0.8 This is a moderately challenging recurrence relation problem requiring algebraic manipulation, summation techniques, and pattern recognition. Part (a) involves computing several terms and solving a cubic equation. Part (b) requires recognizing that the telescoping nature simplifies via the recurrence relation definition, which is a non-trivial insight for most students. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
A sequence $t_1, t_2, t_3, t_4, t_5, \ldots$ is given by
$$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$
where $a$ is a non zero constant.
\begin{enumerate}[label=(\alph*)]
\item Given that $\sum_{r=1}^{3} (r^3 + t_r) = 12$, determine the possible values of $a$. [4]
\item Evaluate $\sum_{r=8}^{31} (t_{r+1} - at_r)$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q6 [8]}}