To find the $n^{th}$ term of the series $2, 6, 18, 54, 162$, we first determine if it is an arithmetic or geometric series. Step 1: Find the common ratio. Divide each term by its preceding term: $$ \frac{6}{2} = 3 $$ $$ \frac{18}{6} = 3 $$ $$ \frac{54}{18} = 3 $$ $$ \frac{162}{54} = 3 $$ Since the ratio between consecutive terms is constant, this is a geometric series with a common ratio $r = 3$. Step 2: Identify the first term. The first term of the series is $a = 2$. Step 3: Use the formula for the $n^{th}$ term of a geometric series. The formula for the $n^{th}$ term ($T_n$) of a geometric series is: $$ T_n = a \cdot r^{n-1} $$ Step 4: Substitute the values of $a$ and $r$ into the formula. $$ T_n = 2 \cdot 3^{n-1} $$ Step 5: Compare with the given options. The derived $n^{th}$ term matches option A. The final answer is $\boxed{\text{A}}$.