Visualizing geometric sequences can really help you understand how they work. It gives you a clear picture of how the numbers in the sequence change. Let’s go over two important ways to look at geometric sequences: the explicit formula and the recursive formula.

1. Understanding the Explicit Formula
   The explicit formula for a geometric sequence looks like this:

   $a_n = a_1 \cdot r^{(n - 1)}$

In this formula, $a_1$ is the first number in the sequence and $r$ is the common ratio.

When we visualize this formula, we can plot the first few numbers of the sequence.

For example, if $a_1 = 2$ and $r = 3$, the sequence would be:
2, 6, 18, 54...

If you make a graph with these points, you’ll see that the numbers grow really fast! This shows how quickly the values increase as $n$ gets bigger.

2. Exploring the Recursive Formula
   The recursive formula looks like this:

   $a_n = r \cdot a_{n-1}$

With this formula, you can build the sequence one step at a time.

Visualizing each number like dots connected by arrows helps you understand how $r$ multiplies each term.

For example, starting with $a_1 = 2$ and $r = 3$, you can follow the arrows:
2 → 6 → 18 → 54.

By visualizing both formulas, you’ll gain a better understanding of how the numbers in geometric sequences relate to each other and how they grow!