1. **State the problem:** Robert wants to retire in 2064 with a retirement account worth $1,000,000 in today's money. He will contribute annually from 2024 to 2063 (40 years) and wants to find the annual contribution amount. 2. **Understand the problem:** The $1,000,000 is in today's money, so we need to adjust it for inflation (discount rate 3%) to find the future value needed in 2064. 3. **Calculate the future value (FV) of the retirement goal:** $$FV = PV \times (1 + r)^n$$ where $PV = 1,000,000$, $r = 0.03$, $n = 40$. $$FV = 1,000,000 \times (1.03)^{40}$$ Calculate: $$FV = 1,000,000 \times 3.2620 = 3,262,000$$ So, Robert needs $3,262,000 in 2064 to have $1,000,000 in today's money. 4. **Calculate the annual contribution needed to reach $3,262,000 in 40 years with 5% return:** Use the future value of an ordinary annuity formula: $$FV = P \times \frac{(1 + i)^n - 1}{i}$$ where $P$ is the annual contribution, $i = 0.05$, $n = 40$, and $FV = 3,262,000$. Rearranged to solve for $P$: $$P = \frac{FV \times i}{(1 + i)^n - 1}$$ 5. **Substitute values:** $$P = \frac{3,262,000 \times 0.05}{(1.05)^{40} - 1}$$ Calculate denominator: $$(1.05)^{40} = 7.0401$$ So: $$P = \frac{163,100}{7.0401 - 1} = \frac{163,100}{6.0401}$$ 6. **Calculate $P$:** $$P = 27,000$$ 7. **Answer:** Robert should deposit approximately $27,000 at the end of each year from 2024 to 2063 to reach his retirement goal.