We are given the recursive sequence:

$a_1 = 4$ $a_n = -4 \cdot a_{n-1} \quad \text{for} \quad n \geq 2$

We need to find $a_6$.

Step-by-Step Calculation

1. Start with $a_1$: $a_1 = 4$

2. Use the recursive formula to calculate $a_2$: $a_2 = -4 \cdot a_1 = -4 \cdot 4 = -16$

3. Now calculate $a_3$: $a_3 = -4 \cdot a_2 = -4 \cdot (-16) = 64$

4. Now calculate $a_4$: $a_4 = -4 \cdot a_3 = -4 \cdot 64 = -256$

5. Now calculate $a_5$: $a_5 = -4 \cdot a_4 = -4 \cdot (-256) = 1024$

6. Finally, calculate $a_6$: $a_6 = -4 \cdot a_5 = -4 \cdot 1024 = -4096$

Final Answer:

$a_6 = -4096$

Would you like a deeper explanation or clarification on any part?

Related Questions:

1. What is the general form of the recursive sequence $a_n$?
2. How does the sign pattern of the terms change as $n$ increases?
3. Can you derive a non-recursive formula for $a_n$?
4. What is the sum of the first 6 terms in this sequence?
5. How would the recursive formula change if the multiplier was different from -4?

Tip: In recursive sequences, the previous term is essential for calculating the next, so understanding the pattern in early terms can help predict further terms.