To determine which expression represents Felipe's balance at the end of the year, we need to perform a series of calculations accounting for both the introductory annual percentage rate (APR) and the standard APR after the introductory period, with monthly compounding. Let's break this down step-by-step:
1. Initial Balance: Felipe starts with a balance of \[tex]$3700.
2. Introductory APR (first 4 months):
- The introductory APR is 5.9%.
- Monthly interest rate during this period is \( \frac{5.9\%}{12} = \frac{0.059}{12} \).
3. Compounding for the Introductory Period (first 4 months):
- We need to calculate the balance after these 4 months of compounding.
- The expression to calculate the balance after 4 months is:
\[ \text{Balance after 4 months} = 3700 \times \left(1 + \frac{0.059}{12}\right)^4 \]
- Calculated balance after 4 months: \( \$[/tex] 3773.305 \).
4. Standard APR (remaining 8 months):
- The standard APR is 17.2%.
- Monthly interest rate for the remaining period is [tex]\( \frac{17.2\%}{12} = \frac{0.172}{12} \)[/tex].
5. Compounding for the Remaining Period (next 8 months):
- We now need to calculate the balance after the remaining 8 months starting from the balance obtained after the introductory period.
- The expression to calculate the balance after the 8 months of compounding is:
[tex]\[ \text{Final balance} = 3773.305 \times \left(1 + \frac{0.172}{12}\right)^8 \][/tex]
- Calculated final balance: [tex]\( \$ 4228.317 \)[/tex].
6. Combining Both Periods:
- The overall expression combining both compounding periods is:
[tex]\[ \$ 3700 \times \left(1 + \frac{0.059}{12}\right)^4 \times \left(1 + \frac{0.172}{12}\right)^8 \][/tex]
Thus, the expression that correctly represents Felipe's balance at the end of the year is:
[tex]\[ A. (\$ 3700)\left(1+\frac{0.059}{12}\right)^4\left(1+\frac{0.172}{12}\right)^8 \][/tex]
Therefore, option A is the correct answer.
