Mathematics: MATHEMATICAL INDUCTION Topical Past Paper Questions

[Maximum mark: 7] Use mathematical induction to prove that $ \frac{\mathrm{d}^n}{\mathrm{d}x^n}(xe^{px}) = p^{n-1}(px + n)e^{px} $ for $ n \in \mathbb{Z}^+ $, $ p \in \mathbb{Q} $.

[Maximum mark: 7] Consider the function $ f(x) = x e^{2x} $, where $ x \in \mathbb{R} $. The $ n^{\text{th}} $ derivative of $ f(x) $ is denoted by $ f^{(n)}(x) $. Prove, by mathematical induction, that $ f^{(n)}(x) = (2^n x + n 2^{n-1}) e^{2x} $, $ n \in \mathbb{Z}^+ $.

[Maximum mark: 6] Use mathematical induction to prove that $ \sum_{r=1}^{n} r(r!) = (n+1)! - 1 $, for $ n \in \mathbb{Z}^{+} $.