We are given a geometric sequence $\{a_n\}$ with $a_1 = 4$ and $a_2 = 20$. We need to find the sum of the first $n$ terms, $S_n$.

1. Problem Description

We are given a geometric sequence $\{a_n\}$ with $a_1 = 4$ and $a_2 = 20$. We need to find the sum of the first $n$ terms, $S_n$.

2. Solution Steps

First, we find the common ratio $r$ of the geometric sequence. Since $a_2 = a_1 \cdot r$, we have

$r = \frac{a_2}{a_1} = \frac{20}{4} = 5$

The formula for the sum of the first $n$ terms of a geometric sequence is:

$S_n = \frac{a_1(r^n - 1)}{r - 1}$

Substituting the given values, we have

$S_n = \frac{4(5^n - 1)}{5 - 1} = \frac{4(5^n - 1)}{4} = 5^n - 1$

3. Final Answer

The sum of the first $n$ terms is $S_n = 5^n - 1$. Therefore, the answer is A. $5^n - 1$.