At the time of her grandson's birth, a grandmother deposits \( \$ 14,000 \) in an account that pays \( 3 \% \) compounded monithly. What will be the value of the account at the child's twenty-first birthday, assuming that no other deposits or withdrawals are made during this period? i) Click the icon to view some finance formulas. The value of the account will be \( \$ \square \). (Round to the nearest dollar as needed.)

To find the future value of the account, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed for. For this problem: - \( P = 14000 \) - \( r = 0.03 \) - \( n = 12 \) (monthly compounding) - \( t = 21 \) years Plugging in the values: \[ A = 14000 \left(1 + \frac{0.03}{12}\right)^{12 \times 21} \] Calculate: \[ A = 14000 \left(1 + 0.0025\right)^{252} \] \[ A = 14000 \left(1.0025\right)^{252} \] Using a calculator, we find: \[ A \approx 14000 \times 1.80611 \approx 25285.54 \] Rounding to the nearest dollar, the value of the account will be \( \$25,286 \). Now for some fun facts and tips: Interestingly, the act of making a single deposit and allowing it to grow through compound interest can lead to substantial wealth over time, showcasing the magic of "time" in finance. This concept is often referred to as the “time value of money,” where money available today is worth more than in the future due to its potential earning capacity. A common mistake many people make is to overlook the effect of compounding frequency. Higher compounding frequency, like monthly instead of annually, results in more interest accrued over the same time period, leading to a significantly larger amount by the end — remember, more is always better when it comes to compounding!