The formula for finding the total of a geometric series is really interesting!

A geometric series happens when you take a number and keep multiplying it by the same amount, called the common ratio (let's call it $r$).

Here’s how to find the sum of the first $n$ terms:

1. Start by writing down the sum: $S_n = a_1 + a_1 r + a_1 r^2 + \ldots + a_1 r^{n-1}$
   (where $a_1$ is the first term).

2. Now, multiply everything by $r$:
   $rS_n = a_1 r + a_1 r^2 + a_1 r^3 + \ldots + a_1 r^n$

3. Next, subtract the second equation from the first:
   $S_n - rS_n = a_1 - a_1 r^n$

4. Then, you can factor and rearrange it to get:
   $S_n(1 - r) = a_1(1 - r^n)$

5. Finally, divide by $(1 - r)$:
   $S_n = \frac{a_1(1 - r^n)}{1 - r}$

Let’s look at an example.

If the first term, $a_1$, is 2, the common ratio $r$ is 3, and you're finding the sum of 4 terms ($n = 4$), it would work out like this:
$S_4 = \frac{2(1 - 3^4)}{1 - 3} = \frac{2(1 - 81)}{-2} = 80.$

So, the sum is 80! This shows how the formula helps us understand how numbers grow in a geometric way.