It seems like you're interested in modeling the growth of bacteria using an exponential growth equation. However, the equation you've provided appears to have some formatting or notational issues.
In general, if the bacteria quadruple each day, we can use the following form of the exponential growth equation:
\[ a_n = a_1 \cdot r^{n-1} \]
Where:
- \( a_n \) = number of bacteria on day \( n \)
- \( a_1 \) = initial number of bacteria (at \( n = 1 \))
- \( r \) = growth factor
- \( n \) = number of days
In your case, \( a_1 \) is the initial number of bacteria after day 1, which is 50, and since the bacteria quadruples each day, the growth factor \( r = 4 \).
The equation for this scenario becomes:
\[ a_n = 50 \cdot 4^{n-1} \]
To find the number of bacteria on subsequent days:
-
For day 1 (\( n = 1 \)): \[ a_1 = 50 \cdot 4^{1-1} = 50 \cdot 4^0 = 50 \cdot 1 = 50 \]
-
For day 2 (\( n = 2 \)): \[ a_2 = 50 \cdot 4^{2-1} = 50 \cdot 4^1 = 50 \cdot 4 = 200 \]
-
For day 3 (\( n = 3 \)): \[ a_3 = 50 \cdot 4^{3-1} = 50 \cdot 4^2 = 50 \cdot 16 = 800 \]
-
For day 4 (\( n = 4 \)): \[ a_4 = 50 \cdot 4^{4-1} = 50 \cdot 4^3 = 50 \cdot 64 = 3200 \]
Summary of the bacteria count:
- Day 1: 50
- Day 2: 200
- Day 3: 800
- Day 4: 3200
If you're looking for \( a_n \) after a certain number of days, substitute the appropriate value for \( n \) in the equation.
If there's something specific you're looking for regarding this equation or the growth of the bacteria, please clarify!