| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Standard +0.8 This is a two-part Further Maths induction question requiring matrix multiplication (non-trivial algebraic manipulation) and divisibility proof. While induction is a standard F1 topic, the matrix algebra requires careful bookkeeping across multiple entries, and students must handle the 3^(n-1) factor correctly. Part (ii) is more routine. This is moderately challenging for Further Maths but not exceptional—above average overall difficulty. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
\begin{enumerate}
\item (i) Prove by induction that for $n \in \mathbb { Z } ^ { + }$
\end{enumerate}
$$\left( \begin{array} { r r }
5 & - 1 \\
4 & 1
\end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c }
2 n + 3 & - n \\
4 n & 3 - 2 n
\end{array} \right)$$
(ii) Prove by induction that for $n \in \mathbb { Z } ^ { + }$
$$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$
is divisible by 7
\hfill \mbox{\textit{Edexcel F1 2024 Q10 [10]}}