| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2019 |
| Marks | 9 |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| Difficulty | Standard +0.3 This is a standard two-part induction question. Part (a) involves proving a recurrence relation formula by induction with straightforward algebra. Part (b) requires induction on a summation formula, which is slightly more involved algebraically but follows a completely standard template. Both parts are routine exercises in mathematical induction with no novel problem-solving required, making this slightly easier than average for A-level Further Maths. |
| Spec | 4.01a Mathematical induction: construct proofs |
\begin{enumerate}[label=(\alph*)]
\item Given that $u_{n+1} = 5u_n + 4$, $u_1 = 4$, prove by induction that $u_n = 5^n - 1$. [4]
\item For all positive integers, $n \geq 2$, prove by induction that
$$\sum_{r=2}^{n} r^2(r-1) = \frac{1}{12}n(n-1)(n+1)(3n+2)$$
[5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2019 Q9 [9]}}