To find the $n$th term of a geometric sequence, we use the formula:

$a_n = a_1 \cdot r^{n-1}$

where:

• $a_1$ is the first term of the sequence,
• $r$ is the common ratio,
• $n$ is the term number.

In this sequence:

• The first term, $a_1$, is $-2$.
• To find the common ratio $r$, we divide the second term by the first term:

$r = \frac{4}{-2} = -2$

So, the common ratio $r$ is $-2$. Now we can substitute $a_1 = -2$ and $r = -2$ into the formula for $a_n$:

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

Therefore, the equation to find the $n$th term, $a_n$, is:

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

Would you like further explanation or have any questions?

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1. How do you identify the common ratio in a geometric sequence?
2. Can the common ratio in a geometric sequence be negative?
3. What is the effect of a negative common ratio on the sequence terms?
4. How would the formula change if the first term were positive?
5. How can we determine if a sequence is geometric?

Tip: In a geometric sequence, the terms alternate in sign if the common ratio is negative.