what is geometric series of sequence 6, 18, 54... up to 25 term

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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Learn how to calculate the sum of a geometric series with the sequence 6, 18, 54... up to 25 terms using the geometric series sum formula. This problem involves identifying the first term, common ratio, and applying the formula to find the sum. Suitable for high school students studying geometric sequences and series.

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