| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove inequality: recurrence sequence |
| Difficulty | Standard +0.3 This is a straightforward induction proof with a simple algebraic manipulation in part (a). The inductive step requires showing u_{n+1} > 5 given u_n > 5, which follows directly from substituting into the given recurrence relation and using basic algebra. While it's an FP1 question, it's a standard textbook-style induction problem requiring no novel insight, making it slightly easier than average overall. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division4.01a Mathematical induction: construct proofs |
\begin{enumerate}[label=(\alph*)]
\item Express $\frac{6x + 10}{x + 3}$ in the form $p + \frac{q}{x + 3}$, where $p$ and $q$ are integers to be found. [1]
\end{enumerate}
The sequence of real numbers $u_1, u_2, u_3, ...$ is such that $u_1 = 5.2$ and $u_{n+1} = \frac{6u_n + 10}{u_n + 3}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Prove by induction that $u_n > 5$, for $n \in \mathbb{Z}^+$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q26 [5]}}