| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2023 |
| Session | October |
| Marks | 10 |
| Topic | Geometric Sequences and Series |
| Type | Find N for S_∞ - S_N condition |
| Difficulty | Moderate -0.8 This is a straightforward geometric progression question requiring only standard formulas and routine calculations. Parts (i) and (ii) are direct applications of GP formulas with no problem-solving required. Part (iii) involves algebraic manipulation and logarithms but follows a predictable pattern for 'sum to infinity minus partial sum' questions. The 'show that' removes any challenge from setting up the inequality. Overall, this is easier than average A-level content. |
| 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 |
The first term of a geometric progression is $10$ and the common ratio is $0.8$.
\begin{enumerate}[label=(\roman*)]
\item Find the fourth term. [2]
\item Find the sum of the first $20$ terms, giving your answer correct to $3$ significant figures. [2]
\item The sum of the first $N$ terms is denoted by $S_N$, and the sum to infinity is denoted by $S_\infty$.
Show that the inequality $S_\infty - S_N < 0.01$ can be written as
$$0.8^N < 0.0002,$$
and use logarithms to find the smallest possible value of $N$. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2023 Q9 [10]}}