[tex]1)\\ \\ \dfrac{n!+(n+1)!}{(n-1)!+n!} = \dfrac{15}{4}\Rightarrow \dfrac{(n-1)!\cdot n+(n-1)!\cdot n\cdot (n+1)}{(n-1)!+(n-1)!\cdot n} = \dfrac{15}{4} \Rightarrow\\ \\ \Rightarrow \dfrac{(n-1)!\cdot \Big(n+n(n+1)\Big)}{(n-1)!\cdot (1+n)} = \dfrac{15}{4} \Rightarrow\\ \\ \Rightarrow \dfrac{n+n(n+1)}{n+1} = \dfrac{15}{4}\Rightarrow \dfrac{n}{n+1}+n =\dfrac{15}{4} \\ \Rightarrow \dfrac{3}{4}+3 = \dfrac{15}{4}\Rightarrow \boxed{n = 3 }\\ \\[/tex]
[tex]2)\quad \dfrac{n!+(n+1)!}{(n-1)!}\leq 24 \Rightarrow \dfrac{(n-1)!n+(n-1)!n(n+1)}{(n-1)!}\leq 24\Rightarrow\\ \\ \Rightarrow \dfrac{(n-1)!\Big(n(n+1)\Big)}{(n-1)!} \leq 24 \Rightarrow n(n+1)\leq 24\Rightarrow \boxed{n\in\{1,2,3,4,5\}}[/tex]
