James deposits $200 in a bank account on the last day of each month. He increases the deposit by $10 each month. The payments continue for a period of 36 months. What is the accumulated value of these payments on the day of the last payment (to the nearest dollar)? The interest rate is 6% p.a. compounded quarterly.
Question
James deposits 10 each month. The payments continue for a period of 36 months. What is the accumulated value of these payments on the day of the last payment (to the nearest dollar)? The interest rate is 6% p.a. compounded quarterly.
Solution 1
First, we need to understand that this is a problem of increasing annuity. An increasing annuity is a series of payments that increase by a constant amount each period.
The formula for the future value of an increasing annuity is:
FV = P * [(1 + r)^n
Solution 2
This problem involves an increasing annuity, where the payments increase at a constant rate each period. The formula for the future value of an increasing annuity is:
FV = P * [(1 + r)^n - 1] / r - Q * [((1 + r)^n - 1) / r - n] / r
where: FV = future value of the annuity P = initial payment per period (in this case, 10 per month) r = interest rate per period (in this case, 6% per year compounded quarterly, or 0.06/4 per quarter) n = number of periods (in this case, 36 months)
However, since the payments are made monthly and the interest is compounded quarterly, we need to convert the interest rate and the number of periods to monthly terms.
The monthly interest rate is (1 + 0.06/4)^(4/12) - 1 = 0.00486755
The calculation would be:
Step 1: Calculate the future value of the annuity:
FV = 10 * [((1 + 0.00486755)^36 - 1) / 0.00486755 - 36] / 0.00486755
Step 2: Round the result to the nearest dollar.
This will give us the accumulated value of these payments on the day of the last payment.
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