We are given the sequence $b_n = n \cdot (-1)^n$ where $n \geq 1$, and we need to find the product of $b_i$ as $i$ ranges from 1 to 4. In other words, we are looking for the following product:

$\prod_{i=1}^{4} b_i = b_1 \cdot b_2 \cdot b_3 \cdot b_4$

Let's evaluate each term of the product:

1. $b_1 = 1 \cdot (-1)^1 = -1$
2. $b_2 = 2 \cdot (-1)^2 = 2$
3. $b_3 = 3 \cdot (-1)^3 = -3$
4. $b_4 = 4 \cdot (-1)^4 = 4$

Now, multiply these terms together:

$\prod_{i=1}^{4} b_i = (-1) \cdot 2 \cdot (-3) \cdot 4$

First, multiply the numbers step by step:

$(-1) \cdot 2 = -2$

$-2 \cdot (-3) = 6$

$6 \cdot 4 = 24$

Thus, the product is:

$\prod_{i=1}^{4} b_i = 24$

Let me know if you need further clarification! Here's a list of related questions to help expand your understanding:

1. What happens to the product if the sequence includes more terms?
2. How does the alternating sign $(-1)^n$ affect the final product?
3. How can this product be generalized for any value of $n$?
4. How would the product change if we considered the sum instead of the product?
5. What is the value of the product for other values of $n$, like 5 or 6?

Tip: When working with sequences involving powers of -1, remember that the signs alternate depending on whether the exponent is odd or even.