Assume a financial problem involves the following type of information. Compound interest Money is paid back using a one time, lump sum payment Borrowing money Choose the correct formula for solving the problem. The correct formula is: A. \( A=P\left(1+\frac{r}{n}\right)^{n t} \) B. \( A=P(1+r t) \) C. \[ P=\frac{p m t\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}{\left(\frac{r}{n}\right)} \] D. \( I=\mathrm{Prt} \) E. \[ A=\frac{\operatorname{pmt}\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{-}\right)} \]

Given the financial problem involves: 1. **Compound Interest**: Interest calculated on the initial principal and also on the accumulated interest from previous periods. 2. **One-Time Lump Sum Payment**: The entire amount (principal + interest) is paid back in a single payment at the end. 3. **Borrowing Money**: This implies calculating how much needs to be repaid after a certain period with interest. Let's evaluate each option: **A. \( A = P\left(1 + \frac{r}{n}\right)^{nt} \)** - **Explanation**: This is the standard **compound interest formula** where: - \( A \) = the future value of the investment/loan, including interest - \( P \) = the principal investment amount (initial deposit or loan amount) - \( r \) = annual interest rate (decimal) - \( n \) = number of times that interest is compounded per year - \( t \) = time the money is invested or borrowed for, in years - **Relevance**: Directly applicable to calculating the lump sum repayment after compounding. **B. \( A = P(1 + rt) \)** - **Explanation**: This is the **simple interest formula**, not suitable for compound interest scenarios. **C. \[ P = \frac{pmt \cdot t \left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right]}{\frac{r}{n}} \] ** - **Explanation**: This resembles the formula for the **present value of an annuity**, which is not applicable for a one-time lump sum payment. **D. \( I = Prt \)** - **Explanation**: This is another form of the **simple interest formula** and does not account for compounding. **E. \[ A = \frac{\text{pmt} \left[\left(1 + \frac{r}{n}\right)^{nt} - 1\right]}{\frac{r}{n}} \] ** - **Explanation**: This formula is similar to the **future value of a series of payments (annuity)**, not a one-time lump sum. **Conclusion:** Option **A** is the correct formula as it accurately represents the compound interest calculation for a lump sum payment after a certain period. **Answer:** **A.** \( A=P\left(1+\frac{r}{n}\right)^{n t} \)