| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove polynomial divisibility property |
| Difficulty | Standard +0.3 Part (i) is a straightforward difference of two squares expansion requiring basic algebra. Part (ii) requires recognizing that the result from (i) gives (3^n)^2 - 1 = 3^{2n} - 1, then showing 3^n is always odd (so 3^n ± 1 are consecutive even numbers, making their product divisible by 8). This is a standard proof by cases or induction-style argument, slightly above average difficulty due to the divisibility reasoning required, but well within typical C3 scope. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
\begin{enumerate}[label=(\roman*)]
\item Multiply out $(3^n + 1)(3^n - 1)$. [1]
\item Hence prove that if $n$ is a positive integer then $3^{2n} - 1$ is divisible by 8. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q6 [4]}}