Eloise contracts to work for 19 ​days, receiving $ 0.02 the first​ day, $ 0.06 the second​ day, $ 0.18 the third​ day, and so​ on, with each​ day's pay triple that of the previous day. How much will she earn on the last day of the​ contract?

Question

Question content area bottom
Part 1
A.$ 7 comma 748 comma 409.78
$ 7 comma 748 comma 409.78
B.$ 23 comma 245 comma 229.34
$ 23 comma 245 comma 229.34
C.$ 3 comma 874 comma 204.89
$ 3 comma 874 comma 204.89
D.$ 5 comma 242.88
$ 5 comma 242.88

Answer 1

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