| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 6 |
| Topic | Proof by induction |
| Type | Prove summation formula |
| Difficulty | Moderate -0.8 This is a straightforward proof by induction with a summation formula. The base case is trivial, and the inductive step requires only routine algebraic manipulation of the formula (2n+1)² and factoring a cubic expression. While it's a 6-mark question requiring careful algebra, it follows a completely standard template with no conceptual challenges or novel insights required. |
| Spec | 4.01a Mathematical induction: construct proofs |
Prove by induction that, for all positive integers $n$,
$$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$
[6]
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q2 [6]}}