Solve manually using formulas/actuarial symbols:

a.) Xia received an HMO card which allows her to reimburse all of her healthcare expenses. She had healthcare expenses of 100 per week for the first 50 weeks of the year. She sent in claim forms for all of her reimbursable expenses in bulk at the end of the 50th week and received payment at the end of the 51st week, which she deposited in her account on the same day that she received it.

Calculate the difference between the amount actually accumulated in her savings account at year end and the amount that could have been accumulated if Xia sent in her claim forms at the end of each week and received the payment at the end of the following week. Let the annual effective rate of interest = 10.95%.

b.) A perpetuity with annual payments is payable beginning 10 years from now. The first payment is 50. Each annual payment thereafter is increased by 10 until a payment of 150 is reached. Subsequent payments remain level at 150. This perpetuity is purchased by means of 10 annual premiums, with the first premium of P due immediately. Each premium after the first is 105% of the preceding one. The annual effective interest rates are 5% during the first 9 years and 3% thereafter. Calculate P.