Question

Ms. Patterson proposes to give her daughter Claire an allowance of ​$0.05 on the first day of her 12​-day ​vacation, $ 0.10 on the second​ day, $ 0.20 on the third​ day, and so on. Write an equation of a sequence whose terms correspond to​ Claire's potential allowance. Find the allowance Claire would receive on the last day of her vacation.
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Part 1
A.
a Subscript n Baseline equals 0.05 n​; ​ $0.60
B.
a Subscript n Baseline equals 0.05 left parenthesis 2 right parenthesis Superscript n minus 1​; ​ $102.40
C.
a Subscript n Baseline equals 0.05 left parenthesis 2 right parenthesis Superscript n​; ​ $204.80
D.
a Subscript n Baseline equals 0.05 plus 2 Superscript n minus 1​; ​ $2048.05

Answer 1

To determine the equation of the sequence that corresponds to Claire's potential allowance, we observe the pattern of the allowance over the days of her vacation:

- On Day 1: $0.05
- On Day 2: $0.10
- On Day 3: $0.20
- On Day 4: $0.40
- On Day 5: $0.80
- ...

We can see that the allowance doubles each day. Therefore, we can express the allowance on the n-th day as follows:

- \( a_n = 0.05 \times 2^{(n-1)} \)

Using this formula, let's compute the allowance for the last day (Day 12):

\[
a_{12} = 0.05 \times 2^{(12 - 1)} = 0.05 \times 2^{11}
\]

We know:

\[
2^{11} = 2048
\]

Thus:

\[
a_{12} = 0.05 \times 2048 = 102.4
\]

So, the allowance Claire would receive on the last day of her vacation is $102.40.

Now, identifying the correct answer choice:

- A. \( a_n = 0.05n; \) $0.60 - **Not correct.**
- B. \( a_n = 0.05(2)^{n-1}; \) $102.40 - **Correct!**
- C. \( a_n = 0.05(2)^{n}; \) $204.80 - **Not correct.**
- D. \( a_n = 0.05 + 2^{n-1}; \) $2048.05 - **Not correct.**

The correct answer is **B**: \( a_n = 0.05(2)^{n-1}; \) $102.40.