To find out how much will be in Derek's account 30 years from today, we need to break the problem into parts.
- Determine the future value of the annual deposits. Derek deposits $3,859.00 per year for 14 years, starting 4 years from today. So, the deposits will occur at years 4 through 17.
To find the future value (FV) of these deposits, we use the future value of an annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( P \) = annual deposit ($3,859.00)
- \( r \) = annual interest rate (6.00% or 0.06)
- \( n \) = total number of deposits (14 years)
Plugging in the values, we have:
\[ FV = 3859 \times \frac{(1 + 0.06)^{14} - 1}{0.06} \]
Let's compute \( (1 + 0.06)^{14} \):
\[ (1.06)^{14} \approx 2.2522 \]
Now, calculate the future value:
\[ FV = 3859 \times \frac{2.2522 - 1}{0.06} \] \[ FV = 3859 \times \frac{1.2522}{0.06} \] \[ FV = 3859 \times 20.87 \] \[ FV \approx 80,628.13 \]
- Calculate how much this future value (from step 1) will grow from year 17 to year 30. Since the last deposit is made at year 17 and we need to know the account balance at year 30, we need to account for 13 years of growth (years 18 through 30).
Using the future value formula again, we find out how much the balance will grow over these 13 years:
\[ FV = PV \times (1 + r)^t \]
Where:
- \( PV \) = future value from the previous calculation ($80,628.13)
- \( r \) = 6% (0.06)
- \( t \) = number of years for compounding (13 years)
Calculating:
\[ FV = 80,628.13 \times (1.06)^{13} \] \[ (1.06)^{13} \approx 2.2522 \text{ (calculated earlier)} \] \[ FV \approx 80,628.13 \times 2.2522 \approx 181,714.09 \]
Thus, the total amount in the account 30 years from today will be approximately $181,714.09.