If a, start subscript, 1, end subscript, equals, 4*a*1​=4 and a, start subscript, n, end subscript, equals, minus, 4, a, start subscript, n, minus, 1, end subscript*an*​=−4*an*−1​ then find the value of a, start subscript, 6, end subscript*a*6​.

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.

If a_1 = 4 and a_n = -4 * a_{n-1}, then find the value of a_6.

Solve the recursive sequence where a_1 = 4 and a_n = -4 * a_{n-1}. Learn step-by-step how to calculate a_6 and understand the pattern of the sequence. Suitable for students in Grades 9-12.

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