The general exponential equation for a sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
where:
- \( a_n \) is the number of bacteria after \( n \) days,
- \( a_1 \) is the initial amount of bacteria,
- \( r \) is the growth factor,
- \( n \) is the day number.
From the problem, we know that:
- After the first day (\( n = 2 \)), the number of bacteria is 50 (\( a_2 = 50 \)).
- The bacteria quadruples daily, so the growth factor \( r = 4 \).
Using this information, we can find \( a_1 \):
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Substitute the known values into the formula for \( n = 2 \): \[ a_2 = a_1 \cdot r^{2-1} \implies 50 = a_1 \cdot 4^{1} \]
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Simplifying the equation: \[ 50 = a_1 \cdot 4 \]
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To find \( a_1 \), divide both sides by 4: \[ a_1 = \frac{50}{4} = 12.5 \]
Thus, the value that would represent \( a_1 \) is \( 12.5 \).