A charity lottery is offering a grand prize of $$50000$ at the end of each year for $25$ years.

a$)$ What is the total amount the winner will collect?

b$)$ A fund must be set up to create an annuity for the lottery winner. If interest is earned at $8\%$ per year, compounded annually, determine the present value of the fund on the day the winner is announced.

The answer for a$)1250000$

b$)533738\mathrm{.}80$