To find an equation for the sequence \(-1, -2, -4, ...\), we first need to observe the pattern in the terms provided.
- The first term (\(n=1\)) is \(-1\).
- The second term (\(n=2\)) is \(-2\).
- The third term (\(n=3\)) is \(-4\).
If we look at the absolute values of the terms, we have:
- First term: 1
- Second term: 2
- Third term: 4
The absolute values double with each term, so this suggests a geometric progression with a common ratio of 2.
Thus, we can express the absolute value of the \(n\)-th term as: \[ |a_n| = 2^{n-1} \]
Since each term in the original sequence is negative, we can express \(a_n\) as: \[ a_n = -2^{n-1} \]
Therefore, the equation that describes the sequence is: \[ a_n = - (2^{n-1}) \]
In the format you've requested: \[ a_n = -1 \cdot (2)^{n-1} \]