At age 20 when you graduate, you start saving for retirement. If your investment plan pays an APR of 7.0%, compounded monthly, and you want to have $3.5 million when you retire in 45 years, how much should you deposit monthly?

To solve this, we use the **future value of an ordinary annuity formula**, which applies when making regular, consistent payments over time with compound interest.

The formula is:

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

Where:
- $FV$ is the future value you want to have, which is $3.5 million.
- $P$ is the monthly payment (what we are solving for).
- $r$ is the monthly interest rate.
- $n$ is the total number of payments.

### Step 1: Identify the variables.
- The future value ($FV$) is $3,500,000.
- The annual percentage rate (APR) is 7%, which means the monthly interest rate is:

\[
r = \frac{7\%}{12} = \frac{0.07}{12} \approx 0.0058333
\]

- The time period is 45 years, and since payments are made monthly, the total number of payments is:

\[
n = 45 \times 12 = 540
\]

### Step 2: Rearrange the formula to solve for $P$.
Rearranging the formula to find $P$:

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

### Step 3: Substitute the values and calculate.

Let me calculate that for you.You would need to deposit approximately **$922.85** per month to reach your goal of $3.5 million in 45 years, assuming an APR of 7.0%, compounded monthly.

Would you like further details on any part of the calculation?

Here are 5 related questions:
1. How would the monthly deposit change with a different interest rate, say 5%?
2. What if you wanted to retire in 40 years instead of 45?
3. How does compound interest affect long-term investments like retirement savings?
4. What if you started with an initial deposit before making monthly contributions?
5. How does changing the compounding frequency (e.g., quarterly vs. monthly) affect the outcome?

**Tip:** Starting early with consistent deposits allows compound interest to significantly grow your retirement fund over time.

Find out how much you need to deposit monthly to reach $3.5 million in 45 years, with a 7% APR compounded monthly. This problem explores key concepts of compound interest and future value of annuities, suitable for advanced algebra students.

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