In the exponential equation \( a_n = a_1 \cdot r^{n-1} \), the term \( a_1 \) represents the initial amount of bacteria at the start of the observation period, which is the first day.
Given that the bacteria quadruples daily, we can say:
- On day 1 (when \( n = 1 \)), the number of bacteria is \( a_1 \).
- On day 2 (when \( n = 2 \)), the number of bacteria is \( 4 \cdot a_1 \).
- On day 3 (when \( n = 3 \)), the number of bacteria is \( 4^2 \cdot a_1 \).
- And so on...
According to your information, after the first day, there were 50 bacteria. This means that after the first day (which is considered \( n = 2 \) since we start counting \( n \) from 1), we have:
\[ a_2 = a_1 \cdot r^{2-1} \]
We need to determine \( a_1 \) when \( a_2 = 50 \) and since \( r = 4 \) (because the bacteria quadruples):
\[ 50 = a_1 \cdot 4^{1} \]
Thus, we can solve for \( a_1 \):
\[ 50 = a_1 \cdot 4 \]
\[ a_1 = \frac{50}{4} \]
\[ a_1 = 12.5 \]
So, the value of \( a_1 \) is 12.5.