The given sequence is \(-3, 9, -27, \ldots\).
First, let's examine the terms:
- The first term is \(-3\).
- The second term is \(9\).
- The third term is \(-27\).
From this, we can see that each term appears to be alternating in sign and has a common ratio.
To determine the common ratio \( r \):
- From \(-3\) to \(9\): \( \frac{9}{-3} = -3 \)
- From \(9\) to \(-27\): \( \frac{-27}{9} = -3 \)
Thus, the common ratio is \(-3\).
We can express the nth term of a geometric sequence using the formula: \[ a_n = a_1 \cdot r^{n-1} \] where \( a_1 \) is the first term and \( r \) is the common ratio.
Here, we have:
- \( a_1 = -3 \)
- \( r = -3 \)
Substituting these values into the formula gives: \[ a_n = -3 \cdot (-3)^{n-1} \]
Therefore, the equation describing the sequence is: \[ a_n = -3 \cdot (-3)^{n-1} \]
So, in the requested format: \[ a_n = -3 \cdot (-3)^{n-1} \]