To find the equation for the given sequence \(-1, 3, -9, ...\), we first identify the pattern in the terms:
- The first term \(a_1 = -1\)
- The second term \(a_2 = 3\)
- The third term \(a_3 = -9\)
Next, let's express each term in terms of a common base and a factor that varies based on \(n\):
We see that the terms alternate in sign, and the absolute values of the terms appear to be multiplying by \(-3\) for each term:
- From \(-1\) to \(3\): \(-1 \times -3 = 3\)
- From \(3\) to \(-9\): \(3 \times -3 = -9\)
This suggests a pattern where each term is obtained by multiplying the previous term by \(-3\).
The general form of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{n-1} \] where \(a_1\) is the first term and \(r\) is the common ratio.
In our case:
- First term \(a_1 = -1\)
- Common ratio \(r = -3\)
Thus, we can write the equation for the nth term \(a_n\) as: \[ a_n = -1 \cdot (-3)^{n-1} \]
Finally, we can express the final equation as: \[ a_n = -1 \cdot (-3)^{n-1} \]
If simplifying is needed to avoid a complex expression in a negative base, you would typically leave it as is. However, the required form can also be presented as: \[ a_n = (-1)^{n} \cdot 3^{n-1} \]
Thus, following your request, the completed equation in the desired format is: \[ a_n = -1 \cdot (-3)^{n-1} \]