9. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=2, a_{n}=2\left(n+a_{n-1}\right), n=2,3, \cdots$, find the general term formula of the sequence $\left\{a_{n}\right\}$.
For $n \geqslant 2$, let $a_{n}+k n+b=2\left[a_{n-1}+k(n-1)+b\right]=2 a_{n-1}+2 k n-2 k+2 b$
$\Rightarrow a_{n}=2 a_{n-1}+k n-2 k+b$, comparing with $a_{n}=2 a_{n-1}+2 n$ we get
$\left\{\begin{array}{l}k=2, \\ -2 k+b=0\end{array} \Rightarrow\left\{\begin{array}{l}k=2, \\ b=4 .\end{array} \Rightarrow\right.\right.$ The sequence $\left\{a_{n}+2 n+4\right\}$ is a geometric sequence with a common ratio of 2.
And $a_{1}=2 \Rightarrow a_{1}+2 \cdot 1+4=8 \Rightarrow a_{n}+2 n+4=8 \cdot 2^{n-1}=2^{n+2}$
$\Rightarrow a_{n}=2^{n+2}-2 n-4(n=1,2, \cdots)$.