What amount is required to purchase an annuity that pays $\$ 4000$ at the end of each quarter for the first five years and then pays $\$ 2000$ at the beginning of each month for the subsequent 15 years? Assume that the annuity payments are based on a rate of return of $7.5 \%$ compounded quarterly. (Do not round intermediate calculations and round your final answer to 2 decimal places.) The amount \[ \$ \square \]

Step 1 ：The problem involves two different types of annuity: an ordinary annuity and an annuity due. The ordinary annuity pays at the end of each period, while the annuity due pays at the beginning of each period.

Step 2 ：The first part of the problem involves an ordinary annuity that pays $4000 at the end of each quarter for the first five years. The formula for the present value of an ordinary annuity is: \[PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r\] where: \[PV\] is the present value, \[PMT\] is the payment per period, \[r\] is the interest rate per period, and \[n\] is the number of periods.

Step 3 ：The second part of the problem involves an annuity due that pays $2000 at the beginning of each month for the subsequent 15 years. The formula for the present value of an annuity due is: \[PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r \times (1 + r)\]

Step 4 ：The total amount required to purchase the annuity is the sum of the present values of the two annuities.

Step 5 ：Given that the payment per period for the first annuity (PMT1) is $4000, the payment per period for the second annuity (PMT2) is $2000, the annual interest rate (r_annual) is 0.075, the quarterly interest rate (r_quarterly) is 0.01875, the monthly interest rate (r_monthly) is 0.0062499999999999995, the number of quarters (n_quarters) is 20, and the number of months (n_months) is 180, we can calculate the present value of the first annuity (PV1) as 66201.62271393654 and the present value of the second annuity (PV2) as 217095.27152303248.

Step 6 ：Adding these two present values together, we find that the total amount required to purchase the annuity is 283296.894236969.

Step 7 ：Final Answer: The amount required to purchase the annuity is \(\boxed{283296.89}\).