The formula for the sum of a geometric series is shown.

Sn=a1(1−rn)1−r

A geometric series is shown.

 1, 3, 9, 27, ...

What is the value of S10 for this geometric series?

28

150

29,524

88,572


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.

A geometric series is shown. 1, 3, 9, 27, ... What is the value of S10 for this geometric series? 28, 150, 29,524, 88,572

Learn how to find the sum of the first 10 terms of the geometric series 1, 3, 9, 27, ... using the geometric series sum formula. This solution demonstrates key concepts in algebra and exponents, suitable for students in Grades 8-10.

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