To describe the sequence \( 1, 4, 16, \ldots \), we can analyze the terms in relation to their position \( n \):
- For \( n = 1 \): \( a_1 = 1 \)
- For \( n = 2 \): \( a_2 = 4 \)
- For \( n = 3 \): \( a_3 = 16 \)
We can see that the terms are powers of 4:
- \( 1 = 4^0 \)
- \( 4 = 4^1 \)
- \( 16 = 4^2 \)
This indicates that each term can be expressed as \( a_n = 4^{n-1} \).
Thus, the equation to describe the sequence is:
\[ a_n = 4^{n-1} \]