| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Number Theory |
| Type | Modular arithmetic properties |
| Difficulty | Standard +0.8 This is a two-part divisibility proof requiring manipulation of exponential expressions and modular arithmetic. Part (a) needs factoring differences and recognizing 81-16=65≡0(mod 15), while part (b) requires induction or direct modular reasoning. The algebraic manipulation is non-trivial but follows standard Further Maths proof techniques, placing it moderately above average difficulty. |
| Spec | 4.01a Mathematical induction: construct proofs |
Given that $f(n) = 3^{4n} + 2^{4n + 2}$,
\begin{enumerate}[label=(\alph*)]
\item show that, for $k \in \mathbb{Z}^+$, $f(k + 1) - f(k)$ is divisible by 15, [4]
\item prove that, for $n \in \mathbb{Z}^+$, $f (n)$ is divisible by 5. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q41 [7]}}