| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 7 |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Standard +0.3 This is a straightforward geometric sequence problem requiring students to use the property that u₂²=u₁u₃, solve a quadratic for k, then apply the convergence condition |r|<1 to select the correct value, and finally use the standard sum to infinity formula. While it involves multiple steps, each is a standard technique with no novel insight required, making it slightly easier than average. |
| 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 SM 2024 Q10 [7]}}