To determine the value of \( a_1 \) in the exponential equation \( a_n = a_1 \cdot p^{n-1} \), we need to understand the meaning of the variables:
- \( a_n \): the amount of bacteria after \( n \) days.
- \( a_1 \): the initial amount of bacteria.
- \( p \): the growth factor.
- \( n \): the number of days.
According to the problem, the bacteria quadruples every day, which means the growth factor \( p \) is 4 (since quadrupling means multiplying by 4).
From the information given, we know that after 1 day ( \( n = 2 \)), there are 50 bacteria. So we can set up the equation:
\[ a_2 = a_1 \cdot p^{2-1} \] \[ 50 = a_1 \cdot 4^{1} \] \[ 50 = a_1 \cdot 4 \]
Now, to find \( a_1 \):
\[ a_1 = \frac{50}{4} \] \[ a_1 = 12.5 \]
Thus, the value that would represent \( a_1 \) is 12.5.
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