Angela deposits \( \$ 3000 \) into an account with an APR of \( 2.5 \% \) for 7 years. Find the future value of the account if interest is compounded weekly. Round your answer to two decimal places. Formulas Simple Interest \[ \begin{array}{l} I=P r t \\ A=P(1+r t) \\ A=P\left(1+\frac{r}{n}\right)^{n t} \end{array} \] Future Value for Simple Interest Future Value for Compound Interest Future Value for Continuous Compound Interest \[ A=P e^{r} \] Annual Percentage Yield (APY) \[ \mathrm{APY}=\left(1+\frac{r}{n}\right)^{n}-1 \] Answer

To find the future value using compound interest, we will use the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \). Here, \( P \) is the principal amount (\$3000), \( r \) is the annual interest rate (2.5%, or 0.025), \( n \) is the number of times interest is compounded per year (weekly, so 52), and \( t \) is the time in years (7). Plugging in the values: \[ A = 3000 \left(1 + \frac{0.025}{52}\right)^{52 \times 7} \] Calculating it step-by-step: 1. Calculate \( \frac{0.025}{52} \approx 0.000480769 \). 2. Calculate \( 52 \times 7 = 364 \). 3. Compute \( A = 3000 \left(1 + 0.000480769\right)^{364} \). 4. Calculate \( A = 3000 \left(1.000480769\right)^{364} \). Using \( (1.000480769)^{364} \approx 1.1001 \) (using a calculator), \[ A \approx 3000 \times 1.1001 \approx 3300.30 \] So, the future value of the account is approximately \( \boxed{3300.30} \). Now, let's dive into a little extra info about compound interest! Did you know that compounding frequency can greatly affect your investment? The more frequently interest is compounded, the more you earn! For instance, compounding yearly versus weekly can lead to significantly different returns over time. So, if you want that future value to grow, look for high-frequency compounding options! Also, if you're really interested in maximizing your returns, consider the APY, or Annual Percentage Yield. This gives a better picture of your actual earnings compared to just the interest rate, as it accounts for compounding. So next time you open a savings account, be sure to ask about the APY—it's the secret sauce in boosting your savings!