| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 5 |
| Topic | Geometric Sequences and Series |
| Type | Sum of first n terms |
| Difficulty | Standard +0.3 Part (i) is trivial recall of the geometric series formula. Part (ii) is a standard proof by induction with divisibility, requiring routine algebraic manipulation (factoring out 64 from 9^{k+1} - 9^k - 8 = 9·9^k - 9^k - 8 = 8·9^k - 8). This is a textbook-style induction question with no novel insight required, making it slightly easier than average overall. |
| Spec | 1.04g Sigma notation: for sums of series1.04i Geometric sequences: nth term and finite series sum4.01a Mathematical induction: construct proofs |
A series is given by
$$\sum_{r=1}^k 9^{r-1}$$
\begin{enumerate}[label=(\roman*)]
\item Write down a formula for the sum of this series. [1]
\item Prove by induction that $P(n) = 9^n - 8n - 1$ is divisible by 64 if $n$ is a positive integer greater than 1. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q13 [5]}}