| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.8 This is a non-standard induction proof requiring algebraic manipulation to establish a recurrence relation before applying induction. Part (a) demands careful index manipulation to factor out the required form, and part (b) requires recognizing how to use this relation in the inductive step—more sophisticated than routine induction proofs of summation formulas or divisibility with obvious factorizations. |
| Spec | 4.01a Mathematical induction: construct proofs |
The expression $f(n)$ is given by $f(n) = 24^{n+3} + 33^{n+1}$.
**(a) [3 marks]**
Show that $f(k+1) - 16f(k)$ can be expressed in the form $A \cdot 33^k$, where $A$ is an integer.
**(b) [4 marks]**
Prove by induction that $f(n)$ is a multiple of $11$ for all integers $n \geq 1$.4 The expression $\mathrm { f } ( n )$ is given by $\mathrm { f } ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( k + 1 ) - 16 \mathrm { f } ( k )$ can be expressed in the form $A \times 3 ^ { 3 k }$, where $A$ is an integer.
\item Prove by induction that $\mathrm { f } ( n )$ is a multiple of 11 for all integers $n \geqslant 1$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q4 [7]}}