| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| Difficulty | Standard +0.8 This is a two-part Further Maths induction question requiring (i) proving a closed form for a second-order recurrence relation, which demands careful algebraic manipulation of the inductive step with two base cases, and (ii) proving divisibility by 2304 = 2^8 × 3^2, requiring binomial expansion and modular arithmetic insight. Both parts go beyond routine A-level induction, but follow standard Further Maths techniques without requiring exceptional creativity. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item (i) A sequence of numbers is defined by
\end{enumerate}
$$\begin{gathered}
u _ { 1 } = 5 \quad u _ { 2 } = 13 \\
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1
\end{gathered}$$
Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,
$$u _ { n } = 2 ^ { n } + 3 ^ { n }$$
(ii) Prove by induction that for $n \geqslant 2$, where $n \in \mathbb { Z }$,
$$f ( n ) = 7 ^ { 2 n } - 48 n - 1$$
is divisible by 2304\\
\includegraphics[max width=\textwidth, alt={}, center]{864a8956-ead0-4abd-91f4-1caa6d17f5e8-14_106_58_2403_1884}
\hfill \mbox{\textit{Edexcel F1 2015 Q8 [12]}}