https://file.aiquickdraw.com/mathbot/file/1727049730926fmkvclf2.jpg

The problem in the image involves calculating the value of a 401(k) retirement account under various conditions. Let’s break it down step by step:

### Problem Breakdown:
1. **Contributions**: 
   - Between ages 22 and 30, you contribute **$3,000 per year**.
   - Your employer contributes **$1,500 per year**.
   - Total annual contribution = $3,000 + $1,500 = **$4,500** per year.

2. **Interest Rate**: 
   - The interest rate is **7.8% compounded annually**.

3. **Time Period**: 
   - The contributions last for **8 years** (from age 22 to age 30).

### Part (a) Calculation:
We need to calculate the future value of the 401(k) after 8 years of contributions with an interest rate of 7.8%, compounded annually.

The formula for the future value of an annuity (which involves regular contributions) is:

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

Where:
- $P$ = Annual payment ($4,500)
- $r$ = Interest rate per period (7.8% = 0.078)
- $t$ = Number of periods (8 years)

I will now calculate this for you.The value of the 401(k) after 8 years of contributions, with an interest rate of 7.8% compounded annually, is approximately **$47,520**.

Would you like to proceed with parts (b) and (c) of the problem? Here are some relative questions for further exploration:

1. How would the result change if the interest rate was compounded monthly?
2. What if you stopped contributions but allowed the balance to grow until age 65?
3. How much would you have if you contributed for an additional 5 years (until age 35)?
4. How does the employer contribution affect the final amount compared to only personal contributions?
5. What would happen if you increased your contribution by $500 each year?

**Tip**: Regular contributions and compound interest can significantly increase retirement savings over time.

Suppose that between the ages of 22 and 30, you contribute $3000 per year to a 401(k), and your employer contributes $1500 per year on your behalf. The interest rate is 7.8% compounded annually. What is the value of the 401(k) after 8 years?

Calculate the future value of a 401(k) retirement account with $3000 personal contributions and $1500 employer contributions per year over 8 years, compounded annually at 7.8%. Learn how compound interest affects retirement savings.

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