Suppose you invest \( \$ 150 \) a month for 5 years into an account earning \( 8 \% \) compounded monthly. After 5 years, you leave the money, without making additional deposits, in the account for another 30 years. How much will you have in the end?

Alright, I need to figure out how much money will be in the account after 35 years, given the initial investment and the interest rate. Let's break this down step by step. First, the initial investment is \$150 per month for 5 years. Since the interest is compounded monthly, I need to calculate the future value of this monthly investment over 5 years. Then, after 5 years, the money will continue to grow without any additional deposits for another 30 years. I'll need to calculate the future value of the amount after the initial 5 years over the next 30 years. Let's start with the first part: calculating the future value of the monthly investments over 5 years. The formula for the future value of an ordinary annuity is: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] Where: - \( FV \) is the future value - \( P \) is the monthly payment (\$150) - \( r \) is the monthly interest rate - \( n \) is the total number of payments First, I need to find the monthly interest rate. The annual interest rate is 8%, so the monthly rate is: \[ r = \frac{0.08}{12} \approx 0.0066667 \] Next, the total number of payments over 5 years is: \[ n = 5 \times 12 = 60 \] Plugging these values into the formula: \[ FV = 150 \times \left( \frac{(1 + 0.0066667)^{60} - 1}{0.0066667} \right) \] Calculating \( (1 + 0.0066667)^{60} \): \[ (1.0066667)^{60} \approx 1.489846 \] So, \[ FV = 150 \times \left( \frac{1.489846 - 1}{0.0066667} \right) = 150 \times \left( \frac{0.489846}{0.0066667} \right) \approx 150 \times 73.5 \approx \$11,025 \] So, after 5 years, the investment grows to approximately \$11,025. Now, for the next 30 years, the money will continue to grow without any additional deposits. The future value after an additional 30 years can be calculated using the compound interest formula: \[ FV = P \times (1 + r)^n \] Where: - \( P \) is the principal amount (\$11,025) - \( r \) is the monthly interest rate (0.0066667) - \( n \) is the number of periods (30 years × 12 months = 360 months) Plugging in the numbers: \[ FV = 11,025 \times (1 + 0.0066667)^{360} \] Calculating \( (1.0066667)^{360} \): \[ (1.0066667)^{360} \approx 10.0627 \] So, \[ FV = 11,025 \times 10.0627 \approx \$111,000 \] Therefore, after a total of 35 years, the investment will grow to approximately \$111,000. But wait, let me double-check these calculations to ensure accuracy. Maybe I made a mistake in the exponentiation or the interest rate application. Recalculating the compound interest for 30 years: \[ FV = 11,025 \times (1.0066667)^{360} \] Using a calculator for precision: \[ (1.0066667)^{360} \approx 10.0627 \] Multiplying by \$11,025: \[ 11,025 \times 10.0627 \approx \$111,000 \] Yes, the calculations seem correct. So, the final amount after 35 years is approximately \$111,000.