how do you find the explicit formula of geometric sequences

To find the explicit formula of geometric sequences, you'll need to find a formula for the n^th term.

In symbols, the n^th term of a geometric sequence is:  t_n  =  a·r^n-1.

a = first term     and     r = common ratio

To find the common ratio, divide any term by its preceding term.

Example:  2, 6, 18, 54, 162, ...

a = first term = 2

r = common ratio = 6/3 = 2 (this will be the same anywhere you begin:  162/54 = 3, 54/18 = 3, 18/6 = 3, etc.)

So, the explicit formula is:  t_n  =  2·3^n-1

Each explicit formula will have the exponent "n-1".

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.