The general exponential equation is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
where:
- \( a_n \) is the amount of bacteria after \( n \) days,
- \( a_1 \) is the initial amount of bacteria,
- \( r \) is the growth factor (in this case, the bacteria quadruples each day, so \( r = 4 \)),
- \( n \) is the number of days.
From the information given:
- After the first day (\( n = 1 \)), the amount of bacteria is 50.
Since we want to find \( a_1 \) (the initial amount), we can set up the equation for the first day:
\[ a_1 \cdot r^{1-1} = 50 \]
This simplifies to:
\[ a_1 \cdot 4^{0} = 50 \] \[ a_1 \cdot 1 = 50 \] \[ a_1 = 50 \]
Thus, the value of \( a_1 \) is \( 50 \).