A finite geometric series is that where the consecutive terms have a fix ratio.
Call r the ratio, An the nth term and An+1 the next term to n, then:
An+1 / An = r
For example, the geometric series, 1, 3, 9, 27, 81, ..
You can verify that
3 = 1* 3
9 = 3 * 3
27 = 9 * 3
81 = 27 * 3
So, the formula is An+1 = An * r, so if you know the first term you can find any other term.
For example, Find the fith term of a geometric series where the first term is 5 and the second term is 25.
r = secont term / first term = 25 / 5 = 5
=> Fifth term = First term * r * r * r * r = First term * r^4 = 25 * 5^4 = 15,625
From this you can also see this valid formula:
An = A1 * r^ (n-1).
So now you have two equivalent formulas:
An+1 = An * r, which is called recursive formula, and
An = A1 * r ^ (n-1), which is called explicit formula