Standard +0.8 This is a standard mathematical induction proof with a factorial summation, requiring verification of the base case and an inductive step involving algebraic manipulation of factorials. While the algebra requires care (particularly factoring out (k+1)! and simplifying), the structure is routine and the techniques are well-practiced at A-level Further Maths. It's moderately harder than average due to the factorial manipulation but doesn't require novel insight.
Prove by mathematical induction that $\sum_{r=1}^{n} (r \times r!) = (n+1)! - 1$ for all positive integers $n$. [6]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q3 [6]}}