You are graduating from college at the end of this semester and have decided to invest ​$5 comma 000 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 40 years. If you earn 6 percent compounded annually on your investment of ​$5 comma 000 at the end of each​ year, how much will you have when you retire in 40 ​years? How much will you have if you wait 10 years before beginning to save and only make 30 payments into your retirement​ account?  
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Part 1
Click on the table icon to view the FVIFA​ table: LOADING....
When you retire in 40 ​years, you will have ​$
  
enter your response here.  ​(Round to the nearest​ dollar.)
Part 2
If you wait 10 years before beginning to save and only make 30 payments into your retirement​ account, when you retire you will have ​$
  
enter your response here.  ​(Round to the nearest​ dollar.)

This problem is related to calculating the future value of an annuity, which involves recurring payments compounded at a fixed rate over time. The formula used to compute the future value of an annuity is:

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

Where:
- $FV$ = Future Value of the annuity
- $P$ = Payment per period (in this case, \$5,000)
- $r$ = Interest rate per period (6% annually or 0.06)
- $n$ = Number of payments or periods

### Part 1: 40 Years of Investment

Given:
- $P = 5000$
- $r = 0.06$
- $n = 40$ (since you're making 40 payments)

We can use the formula to calculate how much you'll have after 40 years.

### Part 2: 30 Years of Investment (Starting After 10 Years)

If you wait 10 years to start investing, you'll only have 30 payments instead of 40. The new variables are:
- $n = 30$

Let's calculate both parts.

I'll compute it now.### Part 1: If you invest \$5,000 at the end of each year for 40 years, earning 6% compounded annually, you will have approximately **\$773,810** when you retire.

### Part 2: If you wait 10 years before beginning to invest and only make 30 payments into your retirement account, you will have approximately **\$395,291** when you retire.

Let me know if you want further details or have any questions!

Here are 5 related questions to deepen your understanding:
1. How would the future value change if the interest rate was 7% instead of 6%?
2. What if you made monthly contributions instead of yearly ones?
3. How does compounding frequency affect the growth of your investments?
4. What is the difference in total contributions between the 40-year and 30-year cases?
5. How does the future value change if the initial payment amount is increased?

**Tip:** Starting early with investments gives more time for compounding, which can significantly increase your returns.

You are graduating from college at the end of this semester and have decided to invest ​$5,000 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 40 years. If you earn 6 percent compounded annually on your investment of ​$5,000 at the end of each​ year, how much will you have when you retire in 40 ​years? How much will you have if you wait 10 years before beginning to save and only make 30 payments into your retirement​ account?

This problem demonstrates how to calculate the future value of a Roth IRA with $5,000 annual contributions over 40 years at 6% compounded interest. Learn how early investments can grow significantly compared to starting after 10 years. Suitable for college-level students or anyone learning financial mathematics.

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