To find the equation describing the sequence 2, -4, 8, we notice that the first term (\(a_1\)) is 2, the second term (\(a_2\)) is -4, and the third term (\(a_3\)) is 8.
We can observe that the common ratio between consecutive terms is -2:
- \(-4 / 2 = -2\)
- \(8 / -4 = -2\)
This indicates that the sequence is a geometric sequence with a common ratio of -2.
Now, the general formula for a geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{n-1} \]
Where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.
Plugging in the values we have:
- \(a_1 = 2\)
- \(r = -2\)
So the equation becomes:
\[ a_n = 2 \cdot (-2)^{n-1} \]
Thus, the final answer is:
\[ a_n = 2 \cdot (-2)^{n-1} \]