| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove divisibility |
| Difficulty | Standard +0.3 This is a standard two-part induction question from FP1. Part (a) requires routine divisibility proof with straightforward algebraic manipulation to factor out 3. Part (b) is matrix induction requiring matrix multiplication but follows a predictable pattern. Both parts are textbook exercises with no novel insight required, though slightly above average difficulty due to being Further Maths content and requiring careful algebraic handling. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
For $n \in \mathbb{Z}^+$ prove that
\begin{enumerate}[label=(\alph*)]
\item $2^{3n + 2} + 5^{n + 1}$ is divisible by 3, [9]
\item $\begin{pmatrix} -2 & -1 \\ 9 & 4 \end{pmatrix}^n = \begin{pmatrix} 1-3n & -n \\ 9n & 3n+1 \end{pmatrix}$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q8 [16]}}