To determine Horatio’s balance at the end of the year, let’s go through the problem step by step:
1. Initial Principal Balance:
Horatio starts with a balance of [tex]\(\$ 2600\)[/tex].
2. Introductory Period:
- For the first 5 months, the introductory APR is [tex]\(4.3\%\)[/tex].
- Given that the APR is compounded monthly, we need to divide the APR by 12 to get the monthly interest rate:
[tex]\[
\text{Monthly interest rate for introductory APR} = \frac{4.3\%}{12} = \frac{0.043}{12}
\][/tex]
- We then calculate the balance at the end of the 5-month introductory period using the formula for compound interest:
[tex]\[
\text{Balance after 5 months} = \$2600 \left(1 + \frac{0.043}{12}\right)^5
\][/tex]
3. Standard Period:
- After the introductory period, the standard APR of [tex]\(13.7\%\)[/tex] applies for the remaining 7 months.
- Similarly, we calculate the monthly interest rate for the standard APR:
[tex]\[
\text{Monthly interest rate for standard APR} = \frac{13.7\%}{12} = \frac{0.137}{12}
\][/tex]
- We then use the balance obtained after the 5 months as the principal for the next 7 months and again apply the compound interest formula:
[tex]\[
\text{Balance at the end of the year} = \left(\$2600 \left(1 + \frac{0.043}{12}\right)^5\right) \left(1 + \frac{0.137}{12}\right)^7
\][/tex]
4. Combining the calculations:
Combining both periods, we get the final balance at the end of the year:
[tex]\[
\text{Balance at the end of the year} = \$2600 \left(1 + \frac{0.043}{12}\right)^5 \left(1 + \frac{0.137}{12}\right)^7
\][/tex]
5. Conclusion:
Comparing this expression to the choices given in the multiple-choice question, we see that it matches option C.
Therefore, the correct expression that represents Horatio’s balance at the end of the year is:
[tex]\[
C. (\$ 2600)\left(1+\frac{0.043}{12}\right)^5\left(1+\frac{0.137}{12}\right)^7
\][/tex]
Additionally, based on given information, after performing the calculations, we find:
- Balance after introductory period: [tex]\( \$2646.918378986316 \)[/tex]
- Balance at the end of the year: [tex]\( \$2865.8357153477655 \)[/tex]
