2025 NYJC P2 Q3

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is \(\displaystyle \displaystyle\int_1^3 \mathrm{f}(x) \, \mathrm{d}x\).[3]

Hence evaluate \(\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.[4]