To calculate the final amount after 30 years of leaving the money in the account, we need to consider the two phases: the initial investment phase (3 years of regular deposits) and the subsequent growth phase (30 years of compounding without additional deposits).
Step 1: Calculate the future value of the initial investment phase (3 years of regular deposits).
The formula to calculate the future value of a series of regular deposits is given by:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where:
FV = Future Value of the investment
P = Monthly deposit amount
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years of regular deposits
In this case, P = $140, r = 7% = 0.07 (as a decimal), n = 12 (compounded monthly), and t = 3 years.
FV = 140 * [(1 + 0.07/12)^(12*3) - 1] / (0.07/12)
FV = 140 * [(1.00583333333)^36 - 1] / (0.00583333333)
FV = 140 * [1.23335491462 - 1] / 0.00583333333
FV = 140 * 0.23335491462 / 0.00583333333
FV = 23335491462.0 / 5833.33333
FV ≈ $40,007.57
Step 2: Calculate the future value of the growth phase (30 years without additional deposits).
Now, we have $40,007.57 in the account, and we want to calculate how much it will grow after 30 years at the same 7% interest rate compounded monthly.
The formula for calculating the future value of a single investment is given by:
FV = P * (1 + r/n)^(nt)
Where:
FV = Future Value of the investment (after 30 years)
P = Initial principal amount (after 3 years of regular deposits)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years of growth phase (30 years)
In this case, P = $40,007.57, r = 7% = 0.07 (as a decimal), n = 12 (compounded monthly), and t = 30 years.
FV = 40,007.57 * (1 + 0.07/12)^(12*30)
FV = 40,007.57 * (1.00583333333)^360
FV = 40,007.57 * 4.07819612525
FV ≈ $163,729.78
After leaving the money in the account for another 30 years, you will have approximately $163,729.78.