A2 Sequence and Series

I'm really confused. When do we use the ar^n-1 formula and when do we use the ar^n formula for geometric series questions. Is the ar^n formula used when we're given initial values ? Does it really matter which formula you use?

Assuming either is used consistently/correctly it wont matter. Howver some problems are naturally started at n=0 and use ar^n whereas others are naturally started at n=1 and use ar^(n-1). You just have to be careful/clear in how you define the first term so "a" and the last term to give "n".

To elaborate a bit.
It's better to know what the n stands for. I tend to think n as in "the number of terms in the sequence" (apparently some don't agree with this? But stick to one definition and you're good).

The nth term in a geometric sequence is ar^(n-1). As a gut check, n=1 should give a, i.e. the first term in the sequence.
The sum of first n terms in a geometric sequence is a*(r^n - 1)/(r-1). Again, gut check with n=1 should give you the sum of the first 1 term, which is a convoluted way of saying the first term is a.
Note: Beware of where the brackets are.

Moral of the story though, if in doubt, plug in some obvious values of n, and see if the formula yields what you expect.
(in particular, the questions with years often confuse people, so a sanity check is immensely useful)

To drive home mqb's point, there are n+1, not n, terms in the sequence 0, 1, 2, ..., n. Again, if you aren't sure about this, plug in some easy values of n for sanity check.