| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Proof by induction |
| Type | Prove recurrence relation formula |
| Difficulty | Standard +0.8 This is a proof by induction on a recurrence relation requiring algebraic manipulation to verify the inductive step. Students must substitute the formula into the recurrence relation and simplify a complex fraction involving powers of 5, which is more demanding than standard induction proofs on summations or divisibility. The algebraic manipulation is non-trivial but follows a clear structure once the approach is identified. |
| Spec | 4.01a Mathematical induction: construct proofs |
The sequence $u_1, u_2, u_3, \ldots$ is defined by
$$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$
Prove by induction that, for all integers $n \geq 1$,
$$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q5 [6]}}