To find the equation for the sequence \(-1, 4, -16, \ldots\), we first observe the pattern of the terms:
- The first term \(a_1 = -1\)
- The second term \(a_2 = 4\)
- The third term \(a_3 = -16\)
Looking closely at the values of the terms, we note the following:
- The signs alternate, with the first term being negative, the second positive, and the third negative.
- The absolute values of the terms are \(1, 4, 16\), which can be expressed in terms of powers of \(4\):
- \(|a_1| = 1 = 4^0\)
- \(|a_2| = 4 = 4^1\)
- \(|a_3| = 16 = 4^2\)
This suggests that the magnitude of the \(n\)-th term is \(4^{n-1}\).
To capture the alternating signs, we can use the term \((-1)^n\), which will let the terms alternate between negative and positive as follows:
- For \(n=1\): \((-1)^1 = -1\)
- For \(n=2\): \((-1)^2 = 1\)
- For \(n=3\): \((-1)^3 = -1\)
Putting this together, the general term \(a_n\) can be expressed as: \[ a_n = (-1)^n \cdot 4^{n-1} \]
Thus the equation for the sequence is:
\[ a_n = (-1)^n (4)^{n-1} \]