To calculate how much Luis will have in his retirement account at the time of his retirement, we use the formula for compound interest with regular contributions. The total value of the account at retirement will be the sum of the compounded initial amount and the future value of the quarterly contributions.

Formula:

$FV = P(1 + \frac{r}{n})^{nt} + \frac{PMT \cdot \left((1 + \frac{r}{n})^{nt} - 1\right)}{\frac{r}{n}}$

Where:

• $FV$: Future value of the account
• $P$: Initial principal ($180,000)
• $r$: Annual interest rate (3.5% = 0.035)
• $n$: Number of compounding periods per year (4, quarterly)
• $t$: Time in years (25)
• $PMT$: Quarterly contribution ($3,000)

Step 1: Compute the future value of the initial principal:

$FV_{P} = P(1 + \frac{r}{n})^{nt}$ $FV_{P} = 180,000 \cdot (1 + \frac{0.035}{4})^{4 \cdot 25}$ $FV_{P} = 180,000 \cdot (1.00875)^{100}$ Using a calculator: $FV_{P} \approx 180,000 \cdot 2.718402$ $FV_{P} \approx 489,312.36$

Step 2: Compute the future value of the quarterly contributions:

$FV_{PMT} = \frac{PMT \cdot \left((1 + \frac{r}{n})^{nt} - 1\right)}{\frac{r}{n}}$ $FV_{PMT} = \frac{3000 \cdot \left((1 + \frac{0.035}{4})^{4 \cdot 25} - 1\right)}{\frac{0.035}{4}}$ $FV_{PMT} = \frac{3000 \cdot \left((1.00875)^{100} - 1\right)}{0.00875}$ $FV_{PMT} = \frac{3000 \cdot (2.718402 - 1)}{0.00875}$ $FV_{PMT} = \frac{3000 \cdot 1.718402}{0.00875}$ $FV_{PMT} \approx \frac{5155.206}{0.00875}$ $FV_{PMT} \approx 589,743.20$

Step 3: Add the two components:

$FV = FV_{P} + FV_{PMT}$ $FV \approx 489,312.36 + 589,743.20$ $FV \approx 1,079,055.56$

Final Answer:

Luis will have approximately $1,079,055.56 in his account at the time of his retirement.

Would you like a detailed explanation of a particular step or related examples? Here are five follow-up questions to expand this topic:

1. How would the total change if the interest rate were compounded monthly instead of quarterly?
2. What happens if Luis decides to stop the quarterly contributions after 10 years?
3. How does the formula change if the compounding is continuous rather than periodic?
4. What is the total interest earned over 25 years in this scenario?
5. How much would Luis need to contribute quarterly to reach $1.5 million at retirement?

Tip: Always compare retirement options by considering both the interest rate and the frequency of compounding to maximize savings.