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 or common difference. Let's check the ratio between consecutive terms: $\frac{6}{2} = 3$ $\frac{18}{6} = 3$ $\frac{54}{18} = 3$ Since there is a constant ratio, this is a geometric series. Step 2: Identify the first term and the common ratio. The first term is $a = 2$. The common ratio is $r = 3$. 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 the derived formula with the given options. A) $T_n = 2(3)^{(n-1)}$ B) $T_n = 2(4)^{(n-1)}$ C) $T_n = 2(1)^{(n-1)}$ D) $T_n = 2(3)^{(n+1)}$ The derived formula matches option A. The final answer is $\boxed{\text{A}}$.