A geometric series is a sum of the terms of a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio.

Given the sequence $6, 18, 54, \ldots$:

• First term ($a$): 6
• Common ratio ($r$): $\frac{18}{6} = 3$

The sum of the first $n$ terms of a geometric series can be calculated using the formula:

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

where:

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

For this sequence, the sum of the first 25 terms ($S_{25}$) is:

$S_{25} = 6 \times \frac{3^{25} - 1}{3 - 1}$

Now let's compute $S_{25}$.The sum of the first 25 terms of the geometric series $6, 18, 54, \ldots$ is $2,541,865,828,326$.

Would you like further details or have any questions? Here are some related questions to explore:

1. How is the common ratio determined in a geometric sequence?
2. What would be the sum of the first 30 terms of this series?
3. How does changing the common ratio affect the sum of a geometric series?
4. What is the general form of a geometric sequence?
5. Can you find the 25th term of this sequence without summing all the terms?

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