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This problem seems to involve sequences, and the variables \(a_1\) and \(S\) indicate the first term and sum of the series, respectively.

Given:
- \( a_1 = -2 \)
- \( S = -\frac{4}{3} \)

You are likely being asked to find the common ratio (\(r\)) of a geometric series since these variables are typical in such problems.

The sum of the first \(n\) terms of a geometric series is given by the formula:

\[
S_n = \frac{a_1 (1 - r^n)}{1 - r}, \quad \text{for } r \neq 1
\]

If we are dealing with the sum of an infinite geometric series, the formula is:

\[
S = \frac{a_1}{1 - r}, \quad \text{where } |r| < 1
\]

### Step-by-Step Solution:
Using the formula for the sum of an infinite geometric series, we substitute the values of \(S\) and \(a_1\):

\[
-\frac{4}{3} = \frac{-2}{1 - r}
\]

Now, solve for \(r\):

1. Multiply both sides by \(1 - r\):
   \[
   -\frac{4}{3}(1 - r) = -2
   \]

2. Distribute the \(-\frac{4}{3}\):
   \[
   -\frac{4}{3} + \frac{4r}{3} = -2
   \]

3. Add \(\frac{4}{3}\) to both sides:
   \[
   \frac{4r}{3} = -2 + \frac{4}{3}
   \]

4. Simplify the right-hand side:
   \[
   \frac{4r}{3} = -\frac{6}{3} + \frac{4}{3} = -\frac{2}{3}
   \]

5. Multiply both sides by 3:
   \[
   4r = -2
   \]

6. Solve for \(r\):
   \[
   r = -\frac{1}{2}
   \]

Thus, the common ratio \(r = -\frac{1}{2}\), which corresponds to answer choice **C**.

### Conclusion:
The correct answer is **C) -\frac{1}{2}**.

Would you like further details or have any questions?

Here are 5 related questions you might want to explore:
1. How do you find the sum of a geometric series with a finite number of terms?
2. What conditions must hold for the sum of an infinite geometric series to exist?
3. How does the common ratio affect the convergence of a geometric series?
4. What is the difference between arithmetic and geometric sequences?
5. Can geometric series be used in practical real-world applications?

**Tip:** Always check if the series converges before applying the infinite sum formula—this happens only when the absolute value of the common ratio is less than 1.

Given a1 = -2, S = -4/3, find the correct answer for the common ratio (r) from the choices.

Solve for the common ratio (r) in a geometric series where a1 = -2 and S = -4/3. This problem explores geometric series concepts, involving the sum formula for infinite series. Suitable for high school students in Grades 10-12.

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