To find the correct equation that describes this sequence $2, -8, 32, -128, \ldots$, let's analyze the pattern between the terms.

Each term appears to be multiplied by $-4$ to obtain the next term:

1. $2 \times -4 = -8$
2. $-8 \times -4 = 32$
3. $32 \times -4 = -128$

Thus, each term is generated by multiplying the previous term by $-4$.

Solution

The equation that best describes this sequence is: $x_n = -4x_{n-1}$

So, the correct answer is: $x_n = -4x_{n-1}$ (last option).

Would you like further details or have any questions?

Here are five related questions to explore this topic further:

1. How can we determine the general formula for a geometric sequence?
2. What is the difference between an arithmetic and a geometric sequence?
3. How does changing the multiplier affect the sequence’s behavior?
4. Can we calculate the $n$-th term directly without recursive steps?
5. How can we determine if a sequence converges or diverges?

Tip: In geometric sequences, observe the ratio between terms to identify the common multiplier.