We can solve this problem by calculating the future value of Luis's retirement account using two components:

1. The present value of the initial $150,000 in the account, which will earn interest over the next 20 years.
2. The future value of the quarterly deposits of $2,000.

1. Future Value of the Initial $150,000

The formula for the future value of a single investment compounded periodically is:

$FV = PV \times (1 + \frac{r}{n})^{nt}$

Where:

• $PV = 150,000$ (initial investment),
• $r = 5.5\% = 0.055$ (annual interest rate),
• $n = 4$ (compounded quarterly),
• $t = 20$ years.

Substituting the values:

$FV_{\text{initial}} = 150,000 \times \left(1 + \frac{0.055}{4}\right)^{4 \times 20}$

2. Future Value of Quarterly Deposits

The future value of periodic payments (annuity) is given by:

$FV_{\text{deposits}} = P \times \left(\frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}\right)$

Where:

• $P = 2000$ (quarterly deposit),
• $r = 0.055$ (annual interest rate),
• $n = 4$ (compounded quarterly),
• $t = 20$ years.

Substituting the values:

$FV_{\text{deposits}} = 2000 \times \left(\frac{(1 + \frac{0.055}{4})^{4 \times 20} - 1}{\frac{0.055}{4}}\right)$

Let's now calculate both future values and add them together:

Step 1: Future Value of Initial Investment

$FV_{\text{initial}} = 150,000 \times \left(1 + \frac{0.055}{4}\right)^{80}$

$FV_{\text{initial}} = 150,000 \times \left(1.01375\right)^{80}$

$FV_{\text{initial}} = 150,000 \times 2.847$

$FV_{\text{initial}} = 427,050$

Step 2: Future Value of Quarterly Deposits

$FV_{\text{deposits}} = 2000 \times \left(\frac{(1.01375)^{80} - 1}{0.01375}\right)$

$FV_{\text{deposits}} = 2000 \times \left(\frac{2.847 - 1}{0.01375}\right)$

$FV_{\text{deposits}} = 2000 \times \left(\frac{1.847}{0.01375}\right)$

$FV_{\text{deposits}} = 2000 \times 134.91$

$FV_{\text{deposits}} = 269,820$

Total Future Value

$FV_{\text{total}} = FV_{\text{initial}} + FV_{\text{deposits}} = 427,050 + 269,820$

$FV_{\text{total}} = 696,870$

Thus, at the time of his retirement, Luis will have $696,870 in his account.

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Would you like more details on how the formulas were applied, or is there another part of the problem you need assistance with?

Here are 5 related questions you might find interesting:

1. How would the future value change if the interest rate increased to 6%?
2. How would the future value change if Luis decided to increase his quarterly deposit to $2,500?
3. What would the future value be if the compounding frequency were annual instead of quarterly?
4. How would the future value differ if Luis started his deposits 5 years later?
5. What is the relationship between the interest rate and the future value of the account?

Tip: Always be mindful of the compounding frequency when using the future value formula—quarterly compounding (4 times a year) can significantly impact the total amount accumulated over time.