10.2 The sequence of numbers $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \ldots$ is formed according to the rule: $\mathrm{x}_{1}=1, \mathrm{x}_{\mathrm{n}+1}=1+\frac{\mathrm{x}_{\mathrm{n}}^{2}}{\mathrm{n}}$ for $\mathrm{n}=1,2,3, \ldots$ Find $\mathrm{x}_{2019}$.
Answer: 2019

Reasoning. $x_{2}=1+\frac{1^{2}}{1}=2, x_{3}=1+\frac{2^{2}}{2}=3$. Hypothesis: $x_{n}=n$. We proceed by induction. If $x_{n}=n$, then $x_{n+1}=1+\frac{n^{2}}{n}=1+n$. The hypothesis is confirmed.