Beginning in January, a person plans to deposit 1000$ at the end of each month into an account earning 6% compounded monthly. Each year taxes must be paid on the interest earned during that year. Find the interest earned during each year for the first 3 years.
To find the interest earned during each year, we first need to determine the balance in the account at the end of each year.
In the first year, there are 12 monthly deposits of $1000, compounded monthly at a 6% annual interest rate. Using the compound interest formula:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.
In this case, P = $0 (since there is no initial balance), r = 6%, n = 12 (compounded monthly), and t = 1 (one year).
So, A = 1000(1 + 0.06/12)^(12*1) = $12,683.16 (rounded to two decimal places).
The interest earned during the first year is the difference between the final balance and the total amount deposited for the year.
Interest earned in the first year = A - (12 * $1000) = $12,683.16 - $12,000 = $683.16.
For the second and third years, we need to consider the additional deposits made each year.
In the second year, there are 12 monthly deposits of $1000, as well as a starting balance of $12,683.16 (the final balance from the first year). Using the same formula:
A = 0(1 + 0.06/12)^(12*0) + 1000(1 + 0.06/12)^(12*1) + 1000(1 + 0.06/12)^(12*2) = $27,227.70 (rounded to two decimal places).
The interest earned during the second year is the difference between the final balance and the total amount deposited for the year.
Interest earned in the second year = A - (12 * $1000) = $27,227.70 - $12,000 = $15,227.70.
Similarly, for the third year:
A = 0(1 + 0.06/12)^(12*0) + 1000(1 + 0.06/12)^(12*1) + 1000(1 + 0.06/12)^(12*2) + 1000(1 + 0.06/12)^(12*3) = $44,104.45 (rounded to two decimal places).
The interest earned during the third year is the difference between the final balance and the total amount deposited for the year.
Interest earned in the third year = A - (12 * $1000) = $44,104.45 - $12,000 = $32,104.45.
Therefore, the interest earned during each of the first three years are as follows:
- First year: $683.16
- Second year: $15,227.70
- Third year: $32,104.45