To describe the sequence \(-3, -6, -12, \ldots\), we first need to determine the relationship between the terms in the sequence.
Let's evaluate how we move from one term to the next:
- The first term (when \(n = 1\)) is \(-3\).
- The second term (when \(n = 2\)) is \(-6\).
- The third term (when \(n = 3\)) is \(-12\).
We can observe that each term is being multiplied by a factor of 2 to get the next term:
- \(-3 \times 2 = -6\)
- \(-6 \times 2 = -12\)
This indicates that we have a geometric sequence with:
- First term \(a = -3\)
- Common ratio \(r = 2\)
The general formula for a geometric sequence is given by: \[ a_n = a \cdot r^{(n-1)} \]
Substituting \(a = -3\) and \(r = 2\): \[ a_n = -3 \cdot 2^{(n-1)} \]
So the equation to describe the sequence is: \[ a_n = -3 \cdot 2^{(n-1)} \] where \(n\) is the position in the sequence, starting from \(n = 1\).