Given the recursive sequence defined by:

$a_1 = 5$ $a_n = -2a_{n-1} \quad \text{for } n > 1$

We need to find the value of $a_4$.

To do this, let's first find the values of $a_2$, $a_3$, and then $a_4$ step by step.

1. Finding $a_2$:

$a_2 = -2a_1 = -2 \times 5 = -10$

2. Finding $a_3$:

$a_3 = -2a_2 = -2 \times (-10) = 20$

3. Finding $a_4$:

$a_4 = -2a_3 = -2 \times 20 = -40$

So, the value of $a_4$ is:

$\boxed{-40}$

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:

1. What is the value of $a_5$ in this sequence?
2. Can you generalize the formula for $a_n$ based on the recursive relation?
3. How does the sign of each term change as $n$ increases?
4. What is the pattern in the absolute values of the sequence terms?
5. How would the sequence change if the recursive formula was $a_n = 2a_{n-1}$ instead?

Tip: When dealing with recursive sequences, calculating the first few terms often reveals a pattern or helps in understanding the nature of the sequence.