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To solve this problem, we need to find the monthly deposit that will accumulate to $500,000 over 25 years, with an account that earns 8% annual interest, compounded monthly.

### Formula for the Future Value of an Annuity:
The formula for the future value of a series of equal monthly payments (ordinary annuity) compounded monthly is given by:

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

Where:
- $FV$ is the future value ($500,000)
- $P$ is the monthly payment (what we are trying to find)
- $r$ is the monthly interest rate (annual rate divided by 12)
- $n$ is the total number of payments (months)

### Given:
- $FV = 500,000$
- Annual interest rate = 8%, so $r = \frac{8\%}{12} = 0.0066667$
- Total time = 25 years, so $n = 25 \times 12 = 300$ months

### Rearranging the formula to solve for $P$:
\[
P = \frac{FV \times r}{(1 + r)^n - 1}
\]

Substitute the values:
\[
P = \frac{500,000 \times 0.0066667}{(1 + 0.0066667)^{300} - 1}
\]

Calculating this will give the required monthly deposit.

Let me calculate it for you.The monthly deposit required to accumulate $500,000 in 25 years, with an 8% annual interest rate compounded monthly, is approximately $525.75.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How does the future value change if the interest rate is increased to 10%?
2. What happens if the deposit period is shortened to 20 years?
3. How would different compounding frequencies (e.g., quarterly or annually) affect the monthly deposit?
4. What would be the total amount contributed by the end of 25 years, and how much of that is interest?
5. How can the present value of a similar annuity be calculated?

**Tip:** When planning for retirement, starting earlier can significantly reduce the amount needed per month due to the power of compound interest.

Suppose you want to have $500,000 for retirement in 25 years. Your account earns 8% interest. How much would you need to deposit in the account each month?

Learn how to calculate the monthly deposit needed to reach a $500,000 retirement goal in 25 years with an 8% interest rate compounded monthly. This problem covers essential concepts such as compound interest, future value of an annuity, and annuities, suitable for students in Grades 10-12.

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