In an arithmetic sequence, the difference between consecutive terms is constant. In formulas, there exists a number such that
In an geometric sequence, the ratio between consecutive terms is constant. In formulas, there exists a number such that
So, there exists infinite sequences that are not arithmetic nor geometric. Simply choose a sequence where neither the difference nor the ratio between consecutive terms is constant.
For example, any sequence starting with
Won't be arithmetic nor geometric. It's not arithmetic (no matter how you continue it, indefinitely), because the difference between the first two numbers is 14, and between the second and the third is -18, and thus it's not constant. It's not geometric either, because the ratio between the first two numbers is 15, and between the second and the third is -1/5, and thus it's not constant.