You want to have $2,350,000 at the end of 23 years. You have $7,000 to invest now, and you will receive $50,000 at the end of 6 years. In addition, you plan to invest an equal amount at the end of every year over the next 23 years to reach your goal. If the annual rate of interest is 6.90%, how much do you have to invest annually

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!

You want to have $2,350,000 at the end of 23 years. You have $7,000 to invest now, and you will receive $50,000 at the end of 6 years. In addition, you plan to invest an equal amount at the end of every year over the next 23 years to reach your goal. If the annual rate of interest is 6.90%, how much do you have to invest annually?

Learn how to calculate the annual investment needed to reach $2,350,000 in 23 years with an initial $7,000 investment and a $50,000 lump sum received after 6 years. Explore the future value formula, compound interest, and annuities at a 6.90% interest rate. Perfect for advanced high school or college-level students.

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