If a, start subscript, 1, end subscript, equals, 5a
1
​
=5 and a, start subscript, n, end subscript, equals, minus, 2, a, start subscript, n, minus, 1, end subscripta
n
​
=−2a
n−1
​
then find the value of a, start subscript, 4, end subscripta
4
​
.

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.

Explore how to find the value of a_4 in a recursive sequence defined by a_1 = 5 and a_n = -2a_{n-1}. Step-by-step solution provided.

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