Start by checking that these sequences really are geometric by dividing each term by its previous term. If this ratio really is constant, we will have found \(r\text{.}\)

1. \(6/3 = 2\text{,}\) \(12/6 = 2\text{,}\) \(24/12 = 2\text{,}\) etc. Yes, to get from any term to the next, we multiply by \(r = 2\text{.}\) So the recursive definition is \(a_n = 2a_{n-1}\) with \(a_0 = 3\text{.}\) The closed formula is \(a_n = 3\cdot 2^{n}\text{.}\)

2. The common ratio is \(r = 1/3\text{.}\) So the sequence has recursive definition \(a_n = \frac{1}{3}a_{n-1}\) with \(a_0 = 27\) and closed formula \(a_n = 27\cdot \frac{1}{3}^{n}\text{.}\)