| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.8 This FP2 question requires algebraic manipulation to show a difference is divisible by 9, then a formal induction proof (non-trivial base case and inductive step), followed by a deductive step using the proven result. While the algebra is manageable, the multi-part structure, proof by induction requirement, and final divisibility deduction make this moderately challenging—harder than routine A-level questions but not requiring exceptional insight. |
| Spec | 4.01a Mathematical induction: construct proofs4.06b Method of differences: telescoping series |
The polynomial $\text{p}(n)$ is given by $\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3$.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $\text{p}(k+1) - \text{p}(k)$, where $k$ is a positive integer, is a multiple of 9. [3 marks]
\item Prove by induction that $\text{p}(n)$ is a multiple of 9 for all integers $n \geqslant 1$. [4 marks]
\end{enumerate}
\item Using the result from part (a)(ii), show that $n(n^2 + 2)$ is a multiple of 3 for any positive integer $n$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q7 [9]}}