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.