Problem: For a sequence u1​,u2​,…, define Δ1(un​)=un+1​−un​ and, for all integers k>1,Δk(un​)=Δ1(Δk−1(un​)). If un​=n3+n, then Δk(un​)=0 for all n
Answer Choices:
A. If k=1
B. If k=2, but not if k=1
C. If k=3, but not if k=2
D. If k=4, but not if k=3
E. For no value of k
Solution:
For all positive integers n,
​Δ1(un​)=(n+1)3+(n+1)−n3−n=3n2+3n+2Δ2(un​)=3(n+1)2+3(n+1)+2−3n2−3n−2=6n+6Δ3(un​)=6Δ4(un​)=0.​