Final answer:
The savings calculation involves two formulas: one for the initial lump sum compounded semiannually and another for monthly deposits treated as an annuity. By applying these, you can ascertain the account's total future value.
Step-by-step explanation:
To calculate the future value of your savings with additional monthly contributions and semiannual compounding interest, you would need to use the future value of an annuity formula for the monthly deposits and a compound interest formula for the initial lump sum. The formula for future value of an annuity (regular deposits) is FV = P \times \left(\frac{\left(1 + r\right)^n - 1}{r}\right), where P is the monthly deposit, r is the monthly interest rate, and n is the total number of deposits. For the initial lump sum, we use the compound interest formula FV = Pv \times \left(1 + r\right)^t, where Pv is the present value, r is the periodic interest rate, and t is the number of periods.
Given a 6% annual interest rate compounded semiannually, we first find the semiannual rate which is 3%. Since we are compounding semiannually but depositing monthly, we should convert the semiannual rate to a monthly rate by dividing it by 6 (as there are two 6-month periods in a year), resulting in a monthly interest rate of 0.5%. Over 20 years, with 12 months in each year, we have a total of 240 deposits. Using these values in the formulas, we can calculate the future value of the savings after 20 years. The initial $50,000 will compound, while the $1,000 monthly contributions will accumulate separately.