https://file.aiquickdraw.com/mathbot/file/1730608279274v322snnw.jpg

Let's solve the problem step by step.

We are given a recursive sequence defined as:
\[
a_1 = -4
\]
\[
a_{n+1} = \frac{3 \, a_n}{n+1}
\]
Our goal is to find the value of \( a_4 \).

### Step-by-Step Calculation

1. **Find \( a_2 \):**
   \[
   a_2 = \frac{3 \, a_1}{1 + 1} = \frac{3 \times (-4)}{2} = \frac{-12}{2} = -6
   \]

2. **Find \( a_3 \):**
   \[
   a_3 = \frac{3 \, a_2}{2 + 1} = \frac{3 \times (-6)}{3} = \frac{-18}{3} = -6
   \]

3. **Find \( a_4 \):**
   \[
   a_4 = \frac{3 \, a_3}{3 + 1} = \frac{3 \times (-6)}{4} = \frac{-18}{4} = -4.5
   \]

### Conclusion
The value of \( a_4 \) is:
\[
a_4 = -4.5
\]

Would you like more details or have questions about the process?

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Here are some related questions to explore:

1. How would the sequence change if \( a_1 \) were positive?
2. What is the behavior of this sequence as \( n \to \infty \)?
3. How does the factor \(\frac{3}{n+1}\) affect the terms of the sequence?
4. Can this recurrence relation be represented by an explicit formula?
5. How do different initial values affect the convergence of the sequence?

**Tip:** When working with recursive sequences, calculating a few terms helps identify patterns or trends.

Considere la sucesión recurrente: a_1 = -4, a_(n+1) = (3 * a_n) / (n + 1). El valor de a_4 es:

Learn how to find the value of a_4 in the recurrence sequence defined by a_1 = -4 and a_(n+1) = (3 * a_n) / (n + 1). Follow a step-by-step solution, covering the concept of recurrence relations and recursive sequence evaluation. Suitable for high school students in Grades 10-12.

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