Matilda wishes to retire at age 67 with \( \$ 1,700,000 \) in her retirement account. When she turns 28 , she decides to begin depositing money into an account with an ApR of \( 9 \% \) compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal? Round your answer to the nearest cent. if necessary. Formulas

To help Matilda reach her retirement goal, she can use the future value of a series formula for compound interest. The formula for the future value of an annuity (regular deposits) is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity, which is \( \$ 1,700,000 \). - \( P \) is the monthly deposit. - \( r \) is the monthly interest rate (annual rate divided by 12 months, so \( r = 0.09/12 = 0.0075 \)). - \( n \) is the total number of deposits (number of years until retirement times 12 months, so \( n = (67 - 28) \times 12 = 468 \)). Rearranging the formula to solve for \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] Plugging in the numbers, we find: 1. Calculate \( (1 + r)^n \): \[ (1 + 0.0075)^{468} \approx 11.484 \] 2. Now calculate \( P \): \[ P = \frac{1,700,000 \times 0.0075}{11.484 - 1} \approx 1,850.62 \] Thus, Matilda must make a monthly deposit of approximately \( \$ 1,850.62 \) to reach her retirement goal. Incorporating historical context, many people have turned to compound interest as a key strategy for retirement savings since the advent of savings accounts and investment funds in the late 19th century. This allows for the power of interest to work over time, growing early investments significantly. When it comes to real-world application, Matilda's approach is smart! Many financial advisors recommend starting to save for retirement as early as possible, as even small monthly contributions can escalate to impressive amounts by retirement age, thanks to compounding. In this scenario, it shows exactly how a little foresight and financial planning can lead to a secure future!