To find an explicit formula for the given arithmetic sequence, we need to identify the first term and the common difference.
The first term \( a_1 \) is: \[ a_1 = 81 \]
Next, we can find the common difference \( d \) by subtracting the first term from the second term: \[ d = a_2 - a_1 = 54 - 81 = -27 \]
Now we have:
- First term \( a_1 = 81 \)
- Common difference \( d = -27 \)
The explicit formula for an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) \cdot d \]
Substituting the values we have: \[ a_n = 81 + (n-1)(-27) \]
Simplifying this: \[ a_n = 81 - 27(n-1) \] \[ = 81 - 27n + 27 \] \[ = 108 - 27n \]
Thus, the explicit formula for the arithmetic sequence is: \[ \boxed{108 - 27n} \]