Let's break down each of the questions:

1. What is a finite geometric series?

   • Correct Answer: A sequence where each term after the first is multiplied by a constant ratio, with a limited number of terms.
     • A geometric series is defined by a constant ratio between consecutive terms, and a finite series has a limited number of terms.

2. If a series has a common ratio of r = 0.5, what happens to the series terms over time?

   • Correct Answer: The terms decrease towards zero.
     • When the common ratio $r$ is less than 1 but positive (e.g., 0.5), the terms gradually decrease toward zero.

3. What is the sum of the first 3 terms if $a = 5$, $r = 3$, and $n = 3$?

   • Correct Answer: 55.
     • The sum $S_n$ of the first $n$ terms of a geometric series is given by $S_n = a \frac{r^n - 1}{r - 1}$. Substituting the given values: $S_3 = 5 \cdot \frac{3^3 - 1}{3 - 1} = 5 \cdot \frac{27 - 1}{2} = 5 \cdot \frac{26}{2} = 5 \cdot 13 = 65.$

4. How can the concept of geometric series apply to technology, specifically in data storage?

   • Correct Answer: By modeling exponential data increase.
     • Geometric series model exponential growth, which applies to the way data storage grows exponentially as more data is added.

5. What role does the formula $S_n = a \frac{1 - r^n}{1 - r}$ play in environmental studies?

   • Correct Answer: It models the accumulation or reduction of substances over time.
     • Geometric series can model how pollutants, resources, or populations change over time.

6. If a geometric series has a positive common ratio less than 1, what is the long-term behavior of the series?

   • Correct Answer: The terms gradually approach zero.
     • With a positive ratio less than 1, the terms of the series get smaller and approach zero.

7. What does the formula $S_n = a \frac{1 - r^n}{1 - r}$ calculate?

   • Correct Answer: The sum of the first $n$ terms of the series.
     • This formula directly calculates the sum of the first $n$ terms of a geometric series.

8. How does understanding geometric series help in environmental management?

   • Correct Answer: By predicting changes in pollutant levels.
     • Geometric series are useful in predicting accumulation or reduction patterns, such as pollutant concentration.

9. Which is a real-world application of geometric series in biology?

   • Correct Answer: Modeling population growth.
     • Population growth is often modeled using exponential functions, which are closely related to geometric series.

10. In the formula $S_n = a \frac{1 - r^n}{1 - r}$, what happens if $n$ approaches infinity and $|r| < 1$?

    • Correct Answer: $S_n$ converges to a finite value.
      • For $|r| < 1$, as $n$ approaches infinity, $r^n$ approaches zero, so the series converges to $S = \frac{a}{1 - r}$.

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the difference between a finite and infinite geometric series?
2. How do you determine if an infinite geometric series converges or diverges?
3. Can a geometric series have a negative common ratio? What would its behavior be?
4. How does the common ratio affect the speed of convergence in a geometric series?
5. How is geometric series used in calculating compound interest?

Tip: Geometric series are key in understanding exponential processes, both in mathematics and real-world applications, like finance, biology, and technology!