2. For the sequence $u_{1}, u_{2}, \cdots$, define $\Delta^{\prime}\left(u_{n}\right)=u_{n+1}-u_{n}$, and for all integers $k>1$, define $\Delta^{k}\left(u_{n}\right)=$ $\Delta^{\prime}\left[\Delta^{k-1}\left(u_{n}\right)\right]$. If $u_{n}=n^{3}+n$, then the condition for $\Delta^{k}\left(u_{n}\right)=0$ for all $n$ is
A. when $k=1$
B. when $k=2$, but not when $k=1$
C. when $k=3$, but not when $k=2$
D. when $k=4$, but not when $k=3$
2. D For all positive integers $n$, we have
$$
\begin{array}{l}
\Delta^{1}\left(u_{n}\right)=\left[(n+1)^{3}+(n+1)\right]-\left(n^{3}+n\right)=3 n^{2}+3 n+2, \\
\Delta^{2}\left(u_{n}\right)=\left[3(n+1)^{2}+3(n+1)+2\right]-\left(3 n^{2}+3 n+2\right)=6 n+6, \\
\Delta^{3}\left(u_{n}\right)=[6(n+1)+6]-(6 n+6)=6, \\
\Delta^{4}\left(u_{n}\right)=0 .
\end{array}
$$

Therefore, the answer is D.