1976 AHSME Problem 25

For a sequence u_1,u_2\ldots , define \Delta^1(u_n)=u_{n+1}-u_n and, for all integer k>1, \Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n)). If u_n=n^3+n, then \Delta^k(u_n)=0 for all n

\begin{array}{l} \textbf{(A) }\text{if }k=1\qquad \\ \textbf{(B) }\text{if }k=2,\text{ but not if }k=1\qquad \\ \textbf{(C) }\text{if }k=3,\text{ but not if }k=2\qquad \\ \textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad\\ \textbf{(E) }\text{for no value of }k \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.