Final answer:
The amount of the quarterly deposits needed to achieve a certain future value with compound interest can be calculated using the future value annuity formula, factoring in the annual interest rate, number of compounding periods per year, and the total number of years.
Step-by-step explanation:
To determine the amount of the quarterly deposits needed to accumulate a future value when interest is compounded quarterly, we use the formula for the future value of an annuity compounded quarterly:
FV = P × { [(1 + r/n)^(nt) - 1] / (r/n) }
where FV is the future value, P is the periodic payment (quarterly deposits), r is the annual interest rate, n is the number of compounds per year (4 for quarterly), and t is the total number of years.
Let's assume an annual interest rate of 10% and the desire to accumulate $10,000 in 10 years. The equation would be set to solve for P as follows:
$10,000 = P × { [(1 + 0.10/4)^(4×10) - 1] / (0.10/4) }
By calculating the above expression, one would find the amount to be deposited quarterly.
To calculate the compound interest specifically, you will need to apply the formula for the compound interest, which includes the principal amount plus the accumulated interest over time. The formula for compound interest is similar to the above but focuses solely on the growth of a single lump sum, not a series of payments.