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 value of the account at the child's twenty-first birthday, 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. ### Given: - \( P = 14000 \) - \( r = 0.03 \) (3% as a decimal) - \( n = 12 \) (compounded monthly) - \( t = 21 \) (the number of years until the child's twenty-first birthday) ### Step-by-step Calculation: 1. **Calculate \( \frac{r}{n} \)**: \[ \frac{r}{n} = \frac{0.03}{12} \] 2. **Calculate \( nt \)**: \[ nt = 12 \times 21 \] 3. **Substitute the values into the formula**: \[ A = 14000 \left(1 + \frac{0.03}{12}\right)^{12 \times 21} \] Now, let's perform the calculations. ### Performing the calculations: 1. Calculate \( \frac{0.03}{12} \): \[ \frac{0.03}{12} = 0.0025 \] 2. Calculate \( 12 \times 21 \): \[ 12 \times 21 = 252 \] 3. Substitute these values into the formula: \[ A = 14000 \left(1 + 0.0025\right)^{252} \] 4. Calculate \( 1 + 0.0025 \): \[ 1 + 0.0025 = 1.0025 \] 5. Now calculate \( (1.0025)^{252} \): \[ (1.0025)^{252} \] 6. Finally, multiply by \( 14000 \): \[ A = 14000 \times (1.0025)^{252} \] Let's perform the final calculations. Calculate the value by following steps: - step0: Calculate: \(14000\times 1.0025^{252}\) - step1: Convert the expressions: \(14000\left(\frac{401}{400}\right)^{252}\) - step2: Simplify: \(14000\times \frac{401^{252}}{400^{252}}\) - step3: Rewrite the expression: \(400\times 35\times \frac{401^{252}}{400^{252}}\) - step4: Reduce the numbers: \(35\times \frac{401^{252}}{400^{251}}\) - step5: Rewrite the expression: \(5\times 7\times \frac{401^{252}}{400^{251}}\) - step6: Rewrite the expression: \(5\times 7\times \frac{401^{252}}{25^{251}\times 16^{251}}\) - step7: Rewrite the expression: \(5\times 7\times \frac{401^{252}}{5^{502}\times 16^{251}}\) - step8: Reduce the numbers: \(7\times \frac{401^{252}}{5^{501}\times 16^{251}}\) - step9: Multiply the fractions: \(\frac{7\times 401^{252}}{5^{501}\times 16^{251}}\) The calculated value of the account at the child's twenty-first birthday is approximately \( 26265.89 \). Rounding to the nearest dollar, the value of the account will be: \[ \text{Value of the account} = \$ 26266 \] Thus, the final answer is: The value of the account will be \( \$ 26266 \).