Question

Is it possible to write a sequence that is both arithmetic and geometric

Answer 1

It is not possible to write a sequence that is both arithmetic and geometric.
An arithmetic sequence is:
a0, a0 + d, a0 + 2d, a0 + 3d, ...

An geometric sequence is:
a0, a0r, a0r2, a0r3, ...

So if the two are equal, we have:
a0 = a0,
a0 + d = a0r,
a0 + 2d = a0r2,
a0 + 3d = a0r3,
...

This can only be true if d = 0 and r =1 and only for the first three terms. It's a trivial sequence with only three elements which can't be necessarily classified as a sequence.