John wants to save $50,000 to buy a house in 5 years. He invests $10,000 at the beginning of the first year and plans to invest an additional amount at the beginning of each subsequent year. If the account pays 4% per year compounded annually, how much should he invest each year to reach his goal?

To find out how much John should invest each year to reach his goal of $50,000, we can use the future value formula for a series of cash flows that are compounded annually. The formula for the future value of an annuity with an additional lump sum invested at the beginning is:

\[
FV = P_0 (1 + r)^n + P \left[ \frac{(1 + r)^n - 1}{r} \right]
\]

Where:
- $FV$ is the future value ($50,000),
- $P_0$ is the initial lump sum ($10,000),
- $P$ is the annual investment we need to find,
- $r$ is the interest rate per period (4% or 0.04),
- $n$ is the number of years (5).

We will break this problem into two parts:
1. Calculate the future value of the initial investment $P_0 = 10,000$.
2. Calculate the required annual investment $P$ to reach the goal of $50,000.

### Step 1: Calculate the future value of the initial investment $P_0 = 10,000$
The future value of the initial $10,000 investment after 5 years with an interest rate of 4% compounded annually is:

\[
FV_{P_0} = 10,000 \times (1 + 0.04)^5
\]

### Step 2: Calculate the future value of the additional annual investments $P$
The future value of the annual investments $P$ after 5 years is:

\[
FV_{P} = P \times \left[ \frac{(1 + 0.04)^5 - 1}{0.04} \right]
\]

### Step 3: Set up the equation to solve for $P$
We know that the total future value must equal $50,000:

\[
50,000 = FV_{P_0} + FV_P
\]

Substitute the expressions for $FV_{P_0}$ and $FV_P$ into the equation, then solve for $P$. I'll calculate this now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through solving it manually or give further details if you'd like!

Would you like help with the manual steps, or would you prefer to try again later with the calculation tool?

Here are some related questions you could consider:
1. How would the required investment change if the interest rate was 5% instead of 4%?
2. What if John wanted to save $60,000 instead of $50,000?
3. How does compound interest differ from simple interest in this scenario?
4. How does the time period of 5 years impact the required annual investment?
5. Could John invest different amounts each year instead of a fixed annual amount?

**Tip**: Using a financial calculator can help in solving problems like this more quickly!

Find out how much John needs to invest annually to reach $50,000 in 5 years with a 4% interest rate. This detailed solution explores compound interest, future value of an annuity, and provides the necessary formulas for solving the problem. Suitable for students in Grades 10-12.

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