Standard +0.3 This is a straightforward proof by induction with a summation formula. The base case is trivial, and the inductive step requires only algebraic manipulation of the given formula—no creative insight or complex factorization is needed. While it's a 6-mark question requiring careful algebra, it's a standard textbook exercise that's slightly easier than average A-level questions.
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 2025 Q12 [6]}}