In the provided exponential equation \( a_n = a_1 \cdot r^{n - 1} \), \( a_n \) represents the amount of bacteria after \( n \) days, \( a_1 \) is the initial amount of bacteria, and \( r \) is the growth factor.
Given that the bacteria quadruples daily, the growth factor \( r = 4 \). According to the problem, after the first day (when \( n = 1 \)), there are 50 bacteria, which means \( a_2 = 50 \).
So, applying the equation for \( n = 2 \):
\[ a_2 = a_1 \cdot r^{n - 1} \]
We can substitute the values:
\[ 50 = a_1 \cdot 4^{2 - 1} \]
This simplifies to:
\[ 50 = a_1 \cdot 4 \]
To find \( a_1 \), divide both sides by 4:
\[ a_1 = \frac{50}{4} = 12.5 \]
Thus, the value of \( a_1 \) is 12.5.
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