To determine how much Eloise will earn on the last day of her 19-day contract, we first need to understand the payment scheme.
The payment each day triples compared to the previous day. We can write the amounts she receives each day based on the first day's payment:
- Day 1: $0.02
- Day 2: $0.02 × 3 = $0.06
- Day 3: $0.06 × 3 = $0.18
- Day 4: $0.18 × 3 = $0.54
- ...
We can see that this can be modeled as a geometric sequence where:
- The first term \(a = 0.02\)
- The common ratio \(r = 3\)
The formula for the \(n\)-th term of a geometric sequence is given by:
\[ a_n = a \cdot r^{(n-1)} \]
To find out how much Eloise will earn on the 19th day, we set \(n=19\):
\[ a_{19} = 0.02 \cdot 3^{(19-1)} = 0.02 \cdot 3^{18} \]
Calculating \(3^{18}\):
\[ 3^{18} = 387420489 \]
Now, substituting this back into the formula:
\[ a_{19} = 0.02 \cdot 387420489 = 7748409.78 \]
Therefore, Eloise will earn $7,748,409.78 on the last day of the contract.
The correct answer is:
A. $7,748,409.78