We know that an is for n = 0, 1, 2, 3, ...
We shall now find out what an is when n is negative.
We know that and we used his to find a0 (we also implicitly used this to find a1).
We shall now use this to find an when n is negative.
We could also have derived this in other ways.
One way is by saying that
a-n = (a-1)n
Another way is by noting that
an x a-n = an - n
Thus a-n x a-n = 1