The problem involves finding a closed form for the sum of a geometric sequence:

$3 + 3^2 + 3^3 + \cdots + 3^n$

Step 1: Recognize it as a geometric series

This is a geometric sequence where:

• The first term $a = 3$.
• The common ratio $r = 3$.

Step 2: Use the sum formula for a geometric series

The sum $S_n$ of the first $n$ terms of a geometric series is given by:

$S_n = a \frac{r^n - 1}{r - 1}$

Here:

• $a = 3$
• $r = 3$

Substituting these values:

$S_n = 3 \cdot \frac{3^n - 1}{3 - 1}$

Simplify:

$S_n = 3 \cdot \frac{3^n - 1}{2}$

Thus, the closed form of the sum is:

$S_n = \frac{3^{n+1} - 3}{2}$

Let me know if you want more detailed steps!