| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 10 |
| Topic | Geometric Sequences and Series |
| Type | Find first term from conditions |
| Difficulty | Standard +0.3 This is a straightforward geometric series problem requiring standard formulas (S∞ = a/(1-r), ar = 18) to find a and r, then algebraic manipulation of the nth term formula and basic logarithm laws. While multi-part with 7 marks total, each step follows routine procedures without requiring problem-solving insight or novel approaches, 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|<11.06f Laws of logarithms: addition, subtraction, power rules |
14.
The sum to infinity of a geometric series is 96\\
The first term of the series is less than 30\\
The second term of the series is 18
\begin{enumerate}[label=(\alph*)]
\item Find the first term and common ratio of the series.\\[0pt]
[4 marks]
\item (i) Show that the $n$th term of the series, $u _ { n }$, can be written as
$$u _ { n } = \frac { 3 ^ { n } } { 2 ^ { 2 n - 5 } }$$
[3 marks]\\
(b) (ii) Hence show that
$$\log _ { 3 } u _ { n } = n \left( 1 - 2 \log _ { 3 } 2 \right) + 5 \log _ { 3 } 2$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q14 [10]}}