A Level Further Maths, Core Pure 1, Chapter 8 · exam-style questions with mark schemes and worked solutions · shadows of past-paper questions and new questions in the style of Edexcel, Edexcel iAL, AQA, OCR A and MEI · proof by mathematical induction: the basis ($n = 1$), the assumption for $n = k$, the inductive step to $n = k + 1$ and the conclusion · sums of series $\sum_{r=1}^{n} f(r)$, divisibility results such as $\mathrm{f}(n) = a^{n} + b^{n}$, powers of matrices $\mathbf{M}^{n}$, sequences given by recurrence relations and $n$th derivatives $\frac{\mathrm{d}^{n}y}{\mathrm{d}x^{n}}$