| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.8 This is a two-part induction question requiring algebraic manipulation to establish a recurrence relation before proving divisibility. Part (a) requires careful expansion and simplification to obtain the form k×15^n, which is non-trivial. Part (b) uses this result in the inductive step, making it more sophisticated than standard divisibility proofs. The combination of algebraic insight and formal proof structure places this above average difficulty for A-level, though it remains accessible to well-prepared Further Maths students. |
| Spec | 4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(n+1) - 8f(n) = 15^{n+1} - 8(15^n - 8^{n-2}) = 15^{n+1} - 8.15^n = 15^n(15-8) = 7.15^n\) | M1A1, M1, A1 | For multiples of powers of 15 only. For valid method ie not using \(120^n\) etc (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(n) \Rightarrow P(n+1)\) and P(2) true | M1, A1, B1, E1 | Or considering \(f(n+1) - f(n)\); \(n=1\) B0; Must score previous 3 marks to be awarded E1 (4 marks) |
**(a)** $f(n+1) - 8f(n) = 15^{n+1} - 8(15^n - 8^{n-2}) = 15^{n+1} - 8.15^n = 15^n(15-8) = 7.15^n$ | M1A1, M1, A1 | For multiples of powers of 15 only. For valid method ie not using $120^n$ etc (4 marks)
**(b)** Assume $f(n)$ is M(7)
Then $f(n+1) - 8f(n) = 7 \times 15^n$
$f(n+1) = M(7) + M(7) = M(7)$
$n=2: f(n) = 15^2 - 8^0 = 224 = 7 \times 32$
$P(n) \Rightarrow P(n+1)$ and P(2) true | M1, A1, B1, E1 | Or considering $f(n+1) - f(n)$; $n=1$ B0; Must score previous 3 marks to be awarded E1 (4 marks)
**Total: 8 marks**
---6
\begin{enumerate}[label=(\alph*)]
\item The function f is given by
$$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$
Express
$$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$
in the form $k \times 15 ^ { n }$.
\item Prove by induction that $15 ^ { n } - 8 ^ { n - 2 }$ is a multiple of 7 for all integers $n \geqslant 2$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q6 [8]}}