Can a sequence be both arithmetic and geometric at the same time

Yes, it can both arithmetic and geometric.

The formula for an arithmetic sequence is: a(n)=a(1)+d(n-1)

The formula for a geometric sequence is: a(n)=a(1)*r^(n-1)

Now, when d is zero and r is one, a sequence is both geometric and arithmetic.

This is because it becomes a(n)=a(1)1 =a(1). Note that a(n) is often written an

It can easily observed that this makes the sequence a constant.