Is it possible that three consecutive elements of an arithmetic sequence with a non-zero difference do not necessarily appear consecutively, but in the same order, in a geometric sequence? Is it possible that they do not necessarily appear in the same order, but directly consecutively?
Both questions can be answered with yes. To prove this, it is enough to provide one example for each case.

In the second case, the numbers $-2, 1, 4$ in this order form an arithmetic sequence with a difference of 3, but when rearranged, the sequence $1, -2, 4$ forms a geometric sequence with a ratio of -2.

For the first case, let's look for numbers in the form $1, q^{2}, q^{3}$; these numbers appear in the geometric sequence $1, q, q^{2}, q^{3}$. These numbers form an arithmetic sequence if $1 + q^{3} = 2q^{2}$. Rearranging and factoring:

$$
(q-1)\left(q^{2}-q-1\right)=0
$$

We see that any root of the equation $q^{2}-q-1=0$, for example, $q=\frac{1+\sqrt{5}}{2}$, is suitable. (The solution $q=1$ is not valid because it would result in a sequence of equal numbers.) Therefore, a possible example for the first question is:

$$
1
$$