| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove summation formula |
| Difficulty | Moderate -0.3 This is a standard proof by induction question with a straightforward algebraic formula. While it requires proper induction structure (base case, assumption, inductive step), the algebra is routine—expanding (2k+1)², factoring cubic expressions, and verifying the formula. It's slightly easier than average because induction proofs of summation formulas are well-practiced exercises with predictable steps, though the algebraic manipulation keeps it from being trivial. |
| Spec | 1.04g Sigma notation: for sums of series4.01a Mathematical induction: construct proofs |
Prove by induction that, for $n \in \mathbb{Z}^+$, $\sum_{r=1}^{n} (2r - 1)^2 = \frac{1}{3} n(2n - 1)(2n + 1)$.
[5]
\hfill \mbox{\textit{Edexcel FP1 Q40 [5]}}