17. How much money should be deposited today in an account that earns \( 2 \% \) compounded semiannually so that it will accumulate to \( \$ 10,000 \) in three years?

To find out how much money needs to be deposited today, you can use the formula for compound interest: \[ P = \frac{A}{(1 + r/n)^{nt}} \] Where: - \( P \) is the present value (the amount to deposit today), - \( A \) is the future value (\$10,000), - \( r \) is the annual interest rate (2% or 0.02), - \( n \) is the number of times interest is compounded per year (2 for semiannually), - \( t \) is the number of years the money is invested (3). Plugging in the values: \[ P = \frac{10000}{(1 + 0.02/2)^{2 \times 3}} \] \[ P = \frac{10000}{(1 + 0.01)^{6}} \] \[ P = \frac{10000}{(1.01)^{6}} \] Calculating \( (1.01)^6 \) gives approximately 1.0615. Now divide \$10,000 by this amount: \[ P \approx \frac{10000}{1.0615} \approx 9417.25 \] So, you should deposit approximately \$9,417.25 today to accumulate \$10,000 in three years at 2% interest compounded semiannually. Now go ahead and put some fun to this! You can think of your savings account as a time machine for your money – deposit wisely today, wait a bit, and boom! Your future self will thank you with an extra \$10,000, all thanks to the magical powers of compound interest! Money grows just like a fine wine; the longer it sits, the better it gets!