Answer all the questions below

.. (1 + 0" .. C. lll'i = 1 __ tq (ax 1) J: +. dam:1 \"AI \"1" . 1th lth Obtain formulas for the evaluation of a life annuity-due to (I) with an initial payment of 1 and with annual payments increasing thereafter by a. 3% of the initial annual payment b. 3% of the previous year' 5 annual payment. Express [Dlxl as an integral and prove the formula 3 _ D ' . = ' . an '5 \"La ea Give an expression for the actuarial accumulated value at age ?0 of an annuity with the following monthly payments: - 100 at the end of each month from age 30 to 40 - 200 at the end of each month from age 40 to 50 - 500 at the end of each month from age 50 to 60 . 1,000 at the end of each month from age 60 to 70. Derive a simplied expression for the actuarial present value for a 25-year term insurance payable immediately on the death of [35), under which the death benet in case of death at age 35 + t is 573, 0 5 t E 25. Interpret your result. Derive a simplied expression for the actuarial present value for an 31-year term insurance payable at the end of the year of death of (I), under which the death benet in case of death in year k + 1 is am 0 5 it