To solve this problem, we need to break it into three parts and calculate the present value and future value of the contributions and investments:

1. Initial Investment: The $7,000 you invest now will grow over the 23 years.
2. Future Investment: The $50,000 you will receive at the end of 6 years will grow from that point to the end of 23 years.
3. Annual Payments: You need to determine the amount to invest annually for 23 years to reach the target of $2,350,000.

The formula we'll use is the Future Value of an Annuity (FVA) and Future Value of a Lump Sum (FV).

Steps to Solve

1. Calculate Future Value of the Initial $7,000 Investment

This grows for 23 years at an annual interest rate of 6.9%. The formula for future value of a lump sum is:

$FV = PV \times (1 + r)^t$

Where:

• $PV = 7000$ (Present value)
• $r = 6.9\% = 0.069$ (Annual interest rate)
• $t = 23$ years

2. Calculate Future Value of the $50,000 received at the end of 6 years

This $50,000 will grow for $23 - 6 = 17$ years. Using the same formula for future value of a lump sum:

$FV = PV \times (1 + r)^t$

Where:

• $PV = 50,000$
• $r = 0.069$
• $t = 17$

3. Future Value of Annual Payments (Annuity)

Let the amount of annual payments be $A$. The formula for the future value of an annuity is:

$FV_{\text{annuity}} = A \times \frac{(1 + r)^t - 1}{r}$

Where:

• $A$ is the annual payment amount we need to find
• $r = 0.069$
• $t = 23$ years

Now, the sum of all these future values should be equal to the goal of $2,350,000.

$FV_{\text{initial}} + FV_{\text{50,000}} + FV_{\text{annuity}} = 2,350,000$

I'll calculate this step-by-step.### Results:

1. Future Value of the Initial $7,000 investment: $32,477.70
2. Future Value of the $50,000 received at the end of 6 years: $155,450.10
3. Required Annual Investment: You need to invest $40,988.04 at the end of each year for the next 23 years to reach your goal of $2,350,000.

Would you like more details or have any other questions? Here are some related questions to consider:

1. How would the required annual investment change if the interest rate were lower?
2. What if you invested more initially instead of $7,000?
3. How does the investment change if you plan to receive the $50,000 after 10 years instead of 6 years?
4. What is the impact of compounding interest on long-term investments?
5. How would the total investment change if the time period was reduced to 20 years?

Tip: The earlier you invest, the more you benefit from compound interest!