| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.8 This is a structured two-part induction proof requiring algebraic manipulation to establish a recurrence relation, then applying it in the inductive step. Part (a) guides students through finding the key relationship (a=7), making the induction more accessible than if unscaffolded. The divisibility proof itself is standard Further Maths fare, but the preliminary algebraic work and the need to recognize how to use the recurrence relation elevates it slightly above a routine induction question. |
| Spec | 4.01a Mathematical induction: construct proofs |
7
\begin{enumerate}[label=(\alph*)]
\item Given that
$$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$
show that
$$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$
where $a$ is an integer.
\item Prove by induction that $12 ^ { n } + 2 \times 5 ^ { n - 1 }$ is divisible by 7 for all integers $n \geqslant 1$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q7 [7]}}