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?

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.