3. Find the largest natural number $\mathrm{n}$, for which the system of inequalities

$1<x<2, 2<x^{2}<3, \ldots, n<x^{n}<n+1$

has a solution. (6 points)
Solution. From the condition $: n=4$.

Rewrite the inequalities as:
$\left\{\begin{aligned} & 16^{3} \text{, then for } n=5 \text{ the given system is already inconsistent: the intervals } [\sqrt[3]{3}, \sqrt[4]{4}] \text{ and } [\sqrt[5]{2}, \sqrt[5]{6}] \text{ do not intersect.} \\ & \text{For } n=4 \text{, it is not difficult to find a value of } x \text{ that satisfies all four inequalities (for example, } x=1.45 \text{).} \end{aligned}\right.$