| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Standard +0.8 This is a Further Maths proof by induction involving matrix powers with a non-trivial result. The algebraic manipulation required in the inductive step—particularly simplifying the (2,1) entry to show (a^(k+1) - b^(k+1))/(a-b)—demands careful factorization and is more sophisticated than standard induction proofs. However, the structure follows a clear template and the matrix multiplication itself is straightforward due to the triangular form. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}
\item Prove by induction that, for $n \in \mathbb { Z } ^ { + }$
\end{enumerate}
$$\left( \begin{array} { l l }
a & 0 \\
1 & b
\end{array} \right) ^ { n } = \left( \begin{array} { c c }
a ^ { n } & 0 \\
\frac { a ^ { n } - b ^ { n } } { a - b } & b ^ { n }
\end{array} \right)$$
where $a$ and $b$ are constants and $a \neq b$.\\
\hfill \mbox{\textit{Edexcel F1 2018 Q8 [5]}}