To determine Clayton's balance at the end of the year given the interest rates and compounding periods, let's break down the question step-by-step.
1. Initial Conditions:
- Initial Balance: \[tex]$4125
- Introductory APR (Annual Percentage Rate): 7.9% for the first 5 months
- Standard APR: 25.7% for the remaining 7 months
- Interest Compounded: Monthly
2. Period Breakdown:
- First 5 months: The APR of 7.9% is in effect.
- Remaining 7 months: The APR of 25.7% is in effect.
3. Monthly Percentage Rates:
- Introductory monthly rate: \( \frac{7.9\%}{12} = \frac{0.079}{12} \)
- Standard monthly rate: \( \frac{25.7\%}{12} = \frac{0.257}{12} \)
4. Calculating balance after the introductory period:
- We need to calculate the balance at the end of 5 months using the introductory rate:
\[
\text{balance\_after\_intro} = 4125 \times \left(1 + \frac{0.079}{12}\right)^5
\]
5. Calculating the final balance after applying the standard APR:
- With the new balance from step 4, we calculate the balance after an additional 7 months using the standard rate:
\[
\text{final\_balance} = \text{balance\_after\_intro} \times \left(1 + \frac{0.257}{12}\right)^7
\]
Following these computations, we find the respective expressions for calculating the balance at each step. The correct expression that encompasses this step-by-step calculation is:
\[
(\$[/tex] 4125)\left(1+\frac{0.079}{12}\right)^{5}\left(1+\frac{0.257}{12}\right)^{7}
\]
Among the given choices, inspect the options carefully. The correct expression that matches our derived formula is:
[tex]\[
\boxed{D. (\$ 4125)\left(1+\frac{0.079}{12}\right)^{5}\left(1+\frac{0.257}{12}\right)^{7}}
\][/tex]
