The timpanist for an orchestra deposits $250 at the beginning of each month in a Roth IRA. The IRA has an annual interest rate of 5.5% compounded monthly. Find the value of the Roth IRA (in dollars) after 12 years.

To find the value of the Roth IRA after 12 years, we can use the formula for the future value of an annuity compounded periodically:

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

Where:
- $ A $ is the future value of the annuity.
- $ P = 250 $ is the monthly deposit.
- $ r = 0.055 $ is the annual interest rate (5.5%).
- $ n = 12 $ is the number of times the interest is compounded per year (monthly).
- $ t = 12 $ is the number of years.

### Steps to calculate:
1. Convert the annual interest rate to the monthly interest rate: $ r/n = 0.055 / 12 $.
2. Use the formula to find the future value of the IRA after 12 years of monthly contributions.

Let's calculate it.The value of the Roth IRA after 12 years is approximately **$50,829.66**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How would the value change if the interest rate increased to 6%?
2. What if the monthly contribution was $300 instead of $250?
3. How does compounding quarterly compare to monthly in this scenario?
4. What if the contributions were made at the end of the month instead of the beginning?
5. How would the value change if the account grew for 15 years instead of 12?

**Tip:** The frequency of compounding significantly impacts the future value of an investment. The more frequent the compounding, the greater the returns!

The timpanist for an orchestra deposits $250 at the beginning of each month in a Roth IRA. The IRA has an annual interest rate of 5.5% compounded monthly. Find the value of the Roth IRA (in dollars) after 12 years.

Calculate the future value of a Roth IRA with monthly deposits of $250 over 12 years at a 5.5% annual interest rate, compounded monthly. Learn about key financial concepts such as annuities and compound interest in this detailed step-by-step solution.

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