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?

To solve this, we will use the **future value of an annuity formula** since you are making monthly deposits into an account that earns interest. The formula is:

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

Where:
- $ FV $ is the future value (which is \$500,000 in this case),
- $ P $ is the monthly deposit (the amount we are trying to find),
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the total number of payments (years multiplied by 12).

### Given:
- $ FV = 500,000 $,
- Annual interest rate = 8%, so $ r = \frac{8\%}{12} = 0.00667 $ (monthly rate),
- Number of years = 25, so $ n = 25 \times 12 = 300 $ months.

Now, we can rearrange the formula to solve for $ P $:

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

Substituting the given values:

\[
P = \frac{500,000 \times 0.00667}{(1 + 0.00667)^{300} - 1}
\]

Let me calculate this.The monthly deposit you would need to make in order to accumulate \$500,000 in 25 years, with an 8% interest rate, is approximately **\$525.75**.

Would you like more details or have any questions about the calculations?

Here are 5 related questions:
1. How would the required monthly deposit change if the interest rate were lower?
2. What if you had only 20 years to save for retirement instead of 25?
3. How does compound interest affect the amount you need to save each month?
4. What would the total amount of interest earned over the 25 years be?
5. What happens if you make an initial lump-sum deposit in addition to the monthly payments?

**Tip:** Small differences in the interest rate or deposit period can greatly affect the total future value due to the power of compound interest.

Calculate how much you need to deposit monthly to reach $500,000 in 25 years at 8% interest. This solution uses the future value of annuities and compound interest formulas, offering a detailed step-by-step breakdown. Suitable for Grades 11-12.

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