1. **State the problem:** Robert wants to have 1,000,000 in his retirement account at the end of 2064. He will contribute every year from 2024 to 2063 (40 years of contributions) and expects a 5% return annually. The discount rate is 3%, but for the calculation of contributions, we focus on the return rate. 2. **Identify the formula:** We use the future value of an ordinary annuity formula since contributions are made at the end of each year: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $FV$ is the future value (1,000,000), - $P$ is the annual payment (what we want to find), - $r$ is the annual interest rate (5% or 0.05), - $n$ is the number of payments (40). 3. **Plug in the values:** $$1,000,000 = P \times \frac{(1 + 0.05)^{40} - 1}{0.05}$$ 4. **Calculate the factor:** $$ (1.05)^{40} = 7.03999$$ So, $$\frac{7.03999 - 1}{0.05} = \frac{6.03999}{0.05} = 120.7998$$ 5. **Solve for $P$:** $$P = \frac{1,000,000}{120.7998}$$ 6. **Simplify:** $$P = 8283.59$$ 7. **Interpretation:** Robert should deposit approximately 8284 every year from 2024 to 2063 to reach 1,000,000 by the end of 2064.