To find an equation that describes the given sequence \( 1, 5, 25, \ldots \), we first need to observe the pattern in the sequence.
Looking at the terms:
- The first term (\( n = 1 \)) is \( 1 = 5^0 \)
- The second term (\( n = 2 \)) is \( 5 = 5^1 \)
- The third term (\( n = 3 \)) is \( 25 = 5^2 \)
From this, we can deduce that the \( n \)-th term of the sequence follows the pattern:
\[ a_n = 5^{n-1} \]
This means that for the first term (\( n = 1 \)), \( a_1 = 5^{1-1} = 5^0 = 1 \), for the second term (\( n = 2 \)), \( a_2 = 5^{2-1} = 5^1 = 5 \), and for the third term (\( n = 3 \)), \( a_3 = 5^{3-1} = 5^2 = 25 \).
Thus, the equation that describes the sequence is:
\[ a_n = 5^{n-1} \]
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