To solve this problem, we will use the formula for the future value of an ordinary annuity, where regular payments are made at the end of each period. The formula is:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value of the annuity (what you want to calculate).
• $P$ is the payment amount per period (in this case, $5,000).
• $r$ is the interest rate per period (here, 8% annually, or 0.08).
• $n$ is the number of payments (in this case, 45 years or 35 years, depending on the scenario).

Part 1: Investment for 45 years

• $P = 5000$ (annual investment),
• $r = 0.08$ (8% annual interest),
• $n = 45$ (45 years of payments).

Plugging the values into the formula:

$FV = 5000 \times \frac{(1 + 0.08)^{45} - 1}{0.08}$

Calculating:

$FV = 5000 \times \frac{(1.08)^{45} - 1}{0.08}$ $FV = 5000 \times \frac{(31.9204) - 1}{0.08}$ $FV = 5000 \times \frac{30.9204}{0.08}$ $FV = 5000 \times 386.505$ $FV \approx 1,932,525.48$

So, the future value if you invest for 45 years will be approximately $1,932,525.48.

Part 2: Wait 10 years, invest for 35 years

Now, you wait 10 years before starting, so you only make 35 payments:

• $P = 5000$,
• $r = 0.08$,
• $n = 35$ (35 years of payments).

Using the formula again:

$FV = 5000 \times \frac{(1 + 0.08)^{35} - 1}{0.08}$

Calculating:

$FV = 5000 \times \frac{(1.08)^{35} - 1}{0.08}$ $FV = 5000 \times \frac{(14.785) - 1}{0.08}$ $FV = 5000 \times \frac{13.785}{0.08}$ $FV = 5000 \times 172.31$ $FV \approx 861,550.28$

So, if you wait 10 years and invest for 35 years, the future value will be approximately $861,550.28.

Summary of Results:

• Investing for 45 years: $1,932,525.48
• Waiting 10 years and investing for 35 years: $861,550.28

Would you like further clarification or details on any part of the solution?

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Here are 5 related questions:

1. What would happen if the interest rate were 6% instead of 8%?
2. How much would you have if you invested $10,000 annually instead of $5,000?
3. What is the difference in future value if the interest is compounded quarterly rather than annually?
4. How much would you need to invest annually to reach $2,000,000 in 45 years?
5. What if you made payments at the beginning of each year instead of at the end?

Tip: Starting your investment early has a huge impact on your retirement savings due to compound interest. The longer your money is invested, the more it grows.