Problem
Which of the following recursive formulas represents the same geometric sequence as the formula $a_{n}=2 \cdot 3^{(n-1)}$ ? A. $\left\{\begin{array}{l}a_{1}=3 \\ a_{n}=a_{n-1} \cdot 2\end{array}\right.$ B. $\left\{\begin{array}{l}a_{1}=2 \\ a_{n}=a_{n-1} \cdot 3\end{array}\right.$ C. $\left\{\begin{array}{l}a_{1}=3 \\ a_{n}=a_{n-1}+2\end{array}\right.$ D. $\left\{\begin{array}{l}a_{1}=2 \\ a_{n}=a_{n-1} \cdot 6\end{array}\right.$
Solution
Step 1 ：The given formula \(a_{n}=2 \cdot 3^{(n-1)}\) represents a geometric sequence where the first term is 2 and the common ratio is 3. We need to find the recursive formula that represents the same sequence.
Step 2 ：Looking at the options, we can see that option B has the first term as 2 and the common ratio as 3, which matches with the given formula.
Step 3 ：So, we can say that option B is the correct answer without any calculations.
Step 4 ：Final Answer: \(\boxed{\text{B. } \left\{\begin{array}{l}a_{1}=2 \\ a_{n}=a_{n-1} \cdot 3\end{array}\right.}\)
