| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 4 |
| Topic | Proof |
| Type | Divisibility proof for all integers |
| Difficulty | Standard +0.8 This is a proof question testing understanding of mathematical induction, requiring students to prove a conditional statement and then demonstrate conceptual understanding of why the inductive step alone is insufficient. The algebraic manipulation is straightforward (showing 8 divides f(n) implies 8 divides f(n+1)), but part (ii) requires genuine insight into the logical structure of induction—recognizing that the base case fails. This goes beyond routine application of induction and tests deeper understanding of proof methodology. |
| Spec | 4.01a Mathematical induction: construct proofs |
Let $f(n) = 2(5^{n-1} + 1)$ for integers $n = 1, 2, 3, \ldots$.
\begin{enumerate}[label=(\roman*)]
\item Prove that, if $f(n)$ is divisible by 8, then $f(n + 1)$ is also divisible by 8. [3]
\item Explain why this result does not imply that the statement
'$f(n)$ is divisible by 8 for all positive integers $n$'
follows by mathematical induction. [1]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q4 [4]}}