Question

True or false: Every sequence is either arithmetic or geometric. If this is true, explain. If false,

Answer 1

In an arithmetic sequence, the difference between consecutive terms is constant. In formulas, there exists a number $r$ such that

$a_{n+1}-a_n=r\quad \forall n\geq 1$

In an geometric sequence, the ratio between consecutive terms is constant. In formulas, there exists a number $r$ such that

$\dfrac{a_{n+1}}{a_n}=r\quad \forall n\geq 1$

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

$1, 15, -3,\ldots$

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.