| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 7 |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Standard +0.8 This question requires students to use the geometric sequence property (u₂² = u₁u₃) to form an equation, solve for k, then apply the convergence condition (|r| < 1) to select the valid solution. Part (b) requires calculating the sum to infinity starting from r=2. While the individual techniques are standard A-level, the combination of algebraic manipulation, quadratic solving, and applying the convergence constraint with justification elevates this slightly above average difficulty. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
The first three terms of a geometric sequence are
$$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$
where $k$ is a constant.
Given that the sequence converges,
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$, giving a reason for your answer.
[4]
\item Find the value of $\sum_{r=2}^{\infty} u_r$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [7]}}