Chapter 5 Polynomials and Polynomial Functions - 5.9 Write Polynomial Functions and Models - 5.9 Exercises - Problem Solving - Page 399: 29

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Take $D_1(n)=R(n+1)-R(n)\\=\frac{(n+1)^3}{3}-(n+1)^2+\frac{8(n+1)}{3}-\frac{n^3}{3}+n^2-\frac{8n}{3}\\=n^2+n+\frac{1}{3}-2n-1+\frac{8}{3}\\=n^2-n+2$ and $D_2(n)=D_(n+1)-D_1(n)\\=(n+1)^2-(n+1)+2-n^2+n-2\\=2n$ $D_3(n)=D_(n+1)-D_2(n)\\=2(n+1)-2n\\=2$ So $D_3(n)=2$ is a constant. Hence, this function has constant third-order differences.