Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

In order to determine if the sequence is geometric, we see if the quotient of all consecutive terms is constant. Here, we have: $\dfrac{c_{n+1}}{c_n}=\dfrac{2(n+1)^3}{2n^3}=\dfrac{(n+1)^3}{n^3}$ This shows that the quotient of all consecutive terms is not constant. Thus, it is not a geometric sequence. In order to determine if the sequence is arithmetic, we see if the difference of all consecutive terms is constant. Here, we have: $c_{n+1}-c_n=2(n+1)^3-2n^3=2(3n^2+3n+1)$ This shows that the difference of all consecutive terms is not constant. Thus, it is not an arithmetic sequence. Hence, the sequence is neither geometric nor arithmetic.