To calculate how much Luis will have in his account at the time of retirement, we use the formula for the future value of an account with periodic payments and a principal amount compounded periodically:

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

Where:

• $P$: Initial principal = $180,000$
• $PMT$: Quarterly contribution = $3,000$
• $r$: Annual interest rate = $0.035$
• $n$: Compounding periods per year = $4$
• $t$: Total number of years = $25$

Step 1: Break down the formula

First term: Contribution from the initial principal ($P$)

$FV_1 = P \cdot \left(1 + \frac{r}{n}\right)^{nt}$

Second term: Contribution from periodic payments ($PMT$)

$FV_2 = \frac{PMT \cdot \left[\left(1 + \frac{r}{n}\right)^{nt} - 1\right]}{\frac{r}{n}}$

Step 2: Compute the values

1. Quarterly interest rate: $\frac{r}{n} = \frac{0.035}{4} = 0.00875$
2. Total compounding periods: $nt = 4 \cdot 25 = 100$

Compute $FV_1$:

$FV_1 = 180,000 \cdot \left(1 + 0.00875\right)^{100}$

Compute $FV_2$:

$FV_2 = \frac{3,000 \cdot \left[\left(1 + 0.00875\right)^{100} - 1\right]}{0.00875}$

Step 3: Add the contributions

Let me calculate this step-by-step.# Define constants P = 180000 # Principal PMT = 3000 # Quarterly contribution r = 0.035 # Annual interest rate n = 4 # Compounding periods per year t = 25 # Total years

Quarterly interest rate and total compounding periods

quarterly_rate = r / n total_periods = n * t

Future value of the initial principal

FV1 = P * (1 + quarterly_rate) ** total_periods

Future value of the quarterly payments

FV2 = PMT * (((1 + quarterly_rate) ** total_periods - 1) / quarterly_rate)

Total future value

total_FV = FV1 + FV2 total_FV906647.3394413196906647.3394413196successThe total amount Luis will have in his retirement account at the time of retirement is approximately $906,647.34.

Would you like details about the calculations, explanations of the formula, or any clarifications? Here are some follow-up questions to explore:

1. How does compounding frequency affect the future value of an account?
2. What happens if Luis adjusts his quarterly contribution?
3. What would be the future value if the interest rate were higher or lower?
4. How does the initial principal contribute to the final amount compared to periodic payments?
5. Can this formula be used for other compounding intervals, such as monthly or annually?

Tip: Regular contributions significantly enhance the future value due to the effect of compounding over time. Adjusting even small contributions can have a big impact!