Acme Annuities recently offered an annuity that pays 3.9% compounded monthly. What equal monthly deposit should be made into this annuity in order to have $159,000 in 14 years?

To solve this, we can use the formula for the future value of an annuity due with periodic payments:

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

Where:
- $ FV $ is the future value (the amount you want to have, $159,000),
- $ P $ is the periodic payment (what we want to find),
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the total number of payments (months in 14 years).

### Step 1: Identify the known values
- Future Value $ FV = 159,000 $
- Annual interest rate $ i = 3.9\% = 0.039 $
- Monthly interest rate $ r = \frac{0.039}{12} = 0.00325 $
- Time period $ t = 14 $ years, or $ n = 14 \times 12 = 168 $ months.

### Step 2: Substitute values into the formula
We rearrange the equation to solve for $ P $:

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

Substitute the known values:

\[
P = \frac{159,000 \times 0.00325}{(1 + 0.00325)^{168} - 1}
\]

Now, let's calculate $ P $.The equal monthly deposit that should be made into the annuity is approximately **$712.95**.

Would you like further clarification on any part of this solution, or would you like to explore related concepts?

Here are 5 related questions to deepen your understanding:
1. How does the formula change if the interest is compounded annually instead of monthly?
2. What happens to the monthly payment if the interest rate increases to 5%?
3. How would the formula apply if the annuity lasts for 20 years instead of 14?
4. Can you derive the formula for the future value of an annuity step by step?
5. How do you calculate the total interest paid over the life of the annuity?

**Tip**: When solving for annuities, always ensure you convert the annual rate to the correct compounding period (monthly, quarterly, etc.).

Learn how to calculate the equal monthly deposit for an annuity offered by Acme Annuities with a 3.9% interest rate compounded monthly. This detailed solution explains the compound interest formula and how to solve for monthly payments over 14 years to achieve a future value of $159,000. Suitable for high school students (Grades 10-12).

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