You are graduating from college at the end of this semester and have decided to invest ​$3 comma 500 at the end of each year into a Roth IRA​ (a retirement investment account that grows tax free and is not taxed when it is​ liquidated) for the next 30 years. If you earn 9 percent compounded annually on your investment of ​$3 comma 500 at the end of each​ year, how much will you have when you retire in 30 ​years? How much will you have if you wait 10 years before beginning to save and only make 20 payments into your retirement​ account?

To solve this problem, we need to use the **Future Value of an Annuity Formula** because the investment is made at the end of each year and it grows with a fixed interest rate.

### Future Value of an Annuity Formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- $ FV $ = Future Value of the annuity
- $ P $ = Payment made each year ($3,500 in this case)
- $ r $ = Interest rate per period (9% = 0.09 in this case)
- $ n $ = Number of periods (years)

### Case 1: You start saving immediately (30 years)

Here:
- $ P = 3500 $
- $ r = 0.09 $
- $ n = 30 $

Let’s calculate the future value for this case.

### Case 2: You wait 10 years before saving (20 years)

Here:
- $ P = 3500 $
- $ r = 0.09 $
- $ n = 20 $

Now let’s calculate both values.### Results:
1. If you invest **$3,500** at the end of each year for **30 years** at a 9% annual interest rate, the total amount you'll have when you retire is approximately **$477,076.38**.
   
2. If you wait **10 years** before starting and invest **$3,500** for **20 years** at the same interest rate, you'll have approximately **$179,060.42** when you retire.

This demonstrates the significant advantage of starting to invest earlier due to the power of compound interest.

Do you want more details or have any questions? 

Here are some related questions:
1. How would the amount change with a different interest rate?
2. What if you invested a higher amount each year, say $5,000?
3. How much would you accumulate if you saved for 40 years instead?
4. How does compound interest impact growth compared to simple interest?
5. What if you made monthly payments instead of annual ones?

**Tip**: The earlier you start investing, the more you benefit from compound interest!

You are graduating from college at the end of this semester and have decided to invest ​$3,500 at the end of each year into a Roth IRA​ (a retirement investment account that grows tax-free and is not taxed when it is​ liquidated) for the next 30 years. If you earn 9 percent compounded annually on your investment of ​$3,500 at the end of each​ year, how much will you have when you retire in 30 ​years? How much will you have if you wait 10 years before beginning to save and only make 20 payments into your retirement​ account?

Learn how to calculate the future value of a Roth IRA investment by contributing $3,500 annually with 9% compounded interest over 30 years. Discover the power of compound interest and compare results when delaying investments by 10 years.

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