\begin{tabular}{l} ch 4 Jack and Jill are saving for a rainy day and decide to put \( \$ 50 \) \\ away in their local bank every year for the next twenty-five years. The \\ local Up-the-Hill Bank will pay them \( 7 \% \) on their account. If Jack and \\ Jill put the money in the account faithfully at the end of every year, how \\ much will they have in it at the end of twenty-five years? (Hint this is an \\ annuity with payments each year, not one tump payment). \\ O\$3162.45 \\ \hline\( \$ 3289.18 \) \\ \hline\( \$ 4122.01 \) \\ \hline \$3987.89 \end{tabular}

To calculate the total amount that Jack and Jill will have in their account at the end of twenty-five years, we can use the formula for the future value of an annuity: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] Where: - \( FV \) is the future value of the annuity - \( P \) is the annual payment - \( r \) is the annual interest rate - \( n \) is the number of years Given: - \( P = \$50 \) (annual payment) - \( r = 7\% = 0.07 \) (annual interest rate) - \( n = 25 \) (number of years) Substitute the values into the formula: \[ FV = 50 \times \left( \frac{(1 + 0.07)^{25} - 1}{0.07} \right) \] Now, we can calculate the future value of the annuity. Calculate the value by following steps: - step0: Calculate: \(\frac{50\left(\left(1+0.07\right)^{25}-1\right)}{0.07}\) - step1: Add the numbers: \(\frac{50\left(1.07^{25}-1\right)}{0.07}\) - step2: Convert the expressions: \(\frac{50\left(\left(\frac{107}{100}\right)^{25}-1\right)}{0.07}\) - step3: Subtract the numbers: \(\frac{50\times \frac{107^{25}-100^{25}}{100^{25}}}{0.07}\) - step4: Multiply the numbers: \(\frac{\frac{107^{25}-100^{25}}{50^{24}\times 2^{25}}}{0.07}\) - step5: Convert the expressions: \(\frac{\frac{107^{25}-100^{25}}{50^{24}\times 2^{25}}}{\frac{7}{100}}\) - step6: Multiply by the reciprocal: \(\frac{107^{25}-100^{25}}{50^{24}\times 2^{25}}\times \frac{100}{7}\) - step7: Rewrite the expression: \(\frac{107^{25}-100^{25}}{50^{24}\times 2^{25}}\times \frac{50\times 2}{7}\) - step8: Reduce the numbers: \(\frac{107^{25}-100^{25}}{50^{23}\times 2^{25}}\times \frac{2}{7}\) - step9: Rewrite the expression: \(\frac{107^{25}-100^{25}}{2^{23}\times 25^{23}\times 2^{25}}\times \frac{2}{7}\) - step10: Reduce the numbers: \(\frac{107^{25}-100^{25}}{2^{22}\times 25^{23}\times 2^{25}}\times \frac{1}{7}\) - step11: Multiply the fractions: \(\frac{107^{25}-100^{25}}{2^{47}\times 25^{23}\times 7}\) - step12: Multiply: \(\frac{107^{25}-100^{25}}{7\times 2^{47}\times 25^{23}}\) The total amount that Jack and Jill will have in their account at the end of twenty-five years is approximately $3162.45.