https://file.aiquickdraw.com/mathbot/file/1726031076035efdc9cvj.jpg

The image shows a formula for the partial sum \( S_n \) of a geometric series. The formula is:

\[
S_n = \sum_{i=1}^{n} T_i = a \frac{1 - r^n}{1 - r}
\]

where:
- \( a \) is the first term of the geometric series,
- \( r \) is the common ratio,
- \( n \) is the number of terms, and
- \( S_n \) is the partial sum up to the \( n \)-th term.

The task presented in the "Quick Check" is to find \( S_4 \), which is the partial sum of the first 4 terms of the geometric series, given certain parameters.

### To solve \( S_4 \):
We would need the values of \( a \) (the first term) and \( r \) (the common ratio). However, these are not visible in the image. Please provide the values of \( a \) and \( r \), and I can calculate the partial sum \( S_4 \).

Would you like further details or have any specific questions? 

Here are 5 follow-up questions:
1. What is the value of the first term \( a \)?
2. What is the common ratio \( r \) in this geometric series?
3. Do you need help understanding how the formula for the sum of a geometric series is derived?
4. Would you like to explore the behavior of geometric series when \( |r| > 1 \)?
5. Do you want to see how this geometric sum formula is applied to different scenarios (e.g., infinite series)?

**Tip**: The geometric sum formula only works when \( r \neq 1 \); otherwise, the series is arithmetic, and a different sum formula is needed.

Find S_4, the partial sum of the first 4 terms of a geometric series, using the formula for the geometric sum provided.

Learn how to calculate the partial sum of a geometric series using the sum formula. This step-by-step guide will show you how to find S_4, the sum of the first 4 terms, including key formulas and concepts in series summation. Suitable for high school students.

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