Let's solve and simplify the given expression for $a_n$:

Given:

$a_n = 4 \cdot (-2)^{n} - 1$

This is a recursive formula where $n$ is the variable, and the expression depends on the power of $(-2)$.

Step-by-step breakdown:

1. Substitute values of $n$: To see how the expression works, we can substitute different values for $n$.

For $n = 1$:

$a_1 = 4 \cdot (-2)^{1} - 1 = 4 \cdot (-2) - 1 = -8 - 1 = -9$ So, $a_1 = -9$.

For $n = 2$:

$a_2 = 4 \cdot (-2)^{2} - 1 = 4 \cdot 4 - 1 = 16 - 1 = 15$ So, $a_2 = 15$.

For $n = 3$:

$a_3 = 4 \cdot (-2)^{3} - 1 = 4 \cdot (-8) - 1 = -32 - 1 = -33$ So, $a_3 = -33$.

For $n = 4$:

$a_4 = 4 \cdot (-2)^{4} - 1 = 4 \cdot 16 - 1 = 64 - 1 = 63$ So, $a_4 = 63$.

2. General Pattern: Based on the results, we see that the values of $a_n$ alternate between negative and positive numbers, depending on whether $n$ is odd or even. Specifically:
   • If $n$ is odd, $a_n$ will be negative.
   • If $n$ is even, $a_n$ will be positive.

Summary of Results:

• For $n = 1$, $a_1 = -9$
• For $n = 2$, $a_2 = 15$
• For $n = 3$, $a_3 = -33$
• For $n = 4$, $a_4 = 63$

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Would you like more details on any part of this process? Feel free to ask.

Further Questions:

1. How can I analyze the behavior of this sequence for large $n$?
2. What is the general formula for this sequence for any given $n$?
3. How would you write a recursive relation for this sequence?
4. Can you use this sequence in real-world applications like population growth or decay?
5. How does the pattern of alternating signs affect the sum of the first $n$ terms?

Tip: When dealing with sequences involving powers of negative numbers, always pay attention to whether the exponent is odd or even, as it determines whether the result will be negative or positive.