To find the sum of the first $n$ terms of a geometric series, we use the formula:

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

where:

• $S_n$ is the sum of the first $n$ terms,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

For the series $1, 3, 9, 27, \ldots$:

• The first term $a_1 = 1$.
• The common ratio $r = \frac{3}{1} = 3$.

We need to find $S_{10}$, the sum of the first 10 terms.

$S_{10} = 1 \cdot \frac{1 - 3^{10}}{1 - 3} = \frac{1 - 3^{10}}{-2}$

Calculating $3^{10}$:

$3^{10} = 59049$

Now substitute it into the formula:

$S_{10} = \frac{1 - 59049}{-2} = \frac{-59048}{-2} = 29524$

So, the value of $S_{10}$ for this geometric series is:

29,524

Would you like more details on this solution or have any questions?

Here are some related questions for further understanding:

1. What is the formula for the sum of an infinite geometric series?
2. How do you determine the common ratio in a geometric sequence?
3. What is the sum of the first 5 terms of the series $1, 3, 9, 27, \ldots$?
4. How does the sum of a geometric series change when the common ratio is less than 1?
5. How do you find the sum of a geometric series with a negative common ratio?

Tip: When using the geometric series sum formula, ensure the common ratio $r$ is not equal to 1, as the formula does not apply in such cases.