a) The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of the six year, decreasing by 1 each year until nothing is paid. Show that the present value is 10 a14 i + a4 i (1 10i) i

b) Find the present value of a perpetuity on which payments are 1 at the end of the 5th and 6th years, 2 at the end of the 7th and 8th years, 3 at the end of the 9th and 10th years, ...,

2. (15 points) Prove and solve the followings (a) (5 points) The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of the six year, decreasing by 1 each year until nothing is paid. Show that the present value is 10 - 077+27: (1 - 101) i (b) (5 points) Find the present value of a perpetuity on which payments are 1 at the end of the 5th and 6th years, 2 at the end of the 7th and 8th years, 3 at the end of the 9th and 10th years, ..., (c) (5 points) Show that ani = -v(Ia)ni 2. (15 points) Prove and solve the followings (a) (5 points) The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of the six year, decreasing by 1 each year until nothing is paid. Show that the present value is 10 - 077+27: (1 - 101) i (b) (5 points) Find the present value of a perpetuity on which payments are 1 at the end of the 5th and 6th years, 2 at the end of the 7th and 8th years, 3 at the end of the 9th and 10th years, ..., (c) (5 points) Show that ani = -v(Ia)ni