The sequence provided in the image is:
$-2, 4, -8, 16, -32$

This is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio. Let's find the common ratio $r$.

$r = \frac{4}{-2} = -2$ $r = \frac{-8}{4} = -2$ $r = \frac{16}{-8} = -2$

So, the common ratio $r$ is $-2$.

The general formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Where:

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

Here, $a_1 = -2$ and $r = -2$. Therefore, the formula for the $n$-th term becomes: $a_n = -2 \cdot (-2)^{n-1}$ Thus, the $n$-th term of the sequence is: $a_n = -2 \cdot (-2)^{n-1}$

Would you like more details on this solution or have any other questions?

Related Questions:

1. How do you determine whether a sequence is arithmetic or geometric?
2. What happens if the common ratio in a geometric sequence is a fraction?
3. Can a sequence have both positive and negative terms and still be geometric?
4. How would the formula change if the first term were positive instead of negative?
5. What is the sum of the first $n$ terms of a geometric sequence?

Tip:

In geometric sequences, the sign of the common ratio determines whether the sequence alternates between positive and negative values.