题目详情 - Q20260129130215989

在$a_n=(n+2)(a_{n+1}-a_n)$中,取$n=1$,可得$a_1=3(a_2-a_1)$,代入$a_2=2$,解得$a_1=\frac{3}{2}$. 又由$a_n=(n+2)(a_{n+1}-a_n)$可得$\frac{a_{n+1}}{a_n}=\frac{n+3}{n+2}$,于是$a_n=\frac{a_n}{a_{n-1}}\cdot\frac{a_{n-1}}{a_{n-2}}\cdots\frac{a_2}{a_1}$. $a_1=\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots\times\frac{n+1}{n}\times\frac{n+2}{n+1}=\frac{n+2}{2}$（$n\geq 2$）,故$a_{2024}=\frac{2024+2}{2}=1\ 013$.故选B.