Is the pattern 1 3 9 27 arithmetic?

Answer 1

No it is not. It is most likely to be geometric

t(n) = 3^(n-1) for n = 1, 2, 3, 4.

But it is equally possible that the sequence is generated by the following order 4 polynomial:

t(n) = (15*n^4 - 118*n^3 + 381*n^2 - 494*n - 240)/24 for n = 1, 2, 3, 4.

Or infinitely many other polynomials.