In each of the special cases h + k = n - 1, and n annual payments are made. The values of h and k are so chosen that the present comes at the desired point - for the amount of an annuity at the time of the last payment; for the present value of an annuity one year before the first payment; for the amount of an annuity-due one year after the last payment; for the present value of an annuity-due at the time of the first payment; and for the present value of a deferred annuity r + 1 years before the first payment.
Proceedings of the Iowa Academy of Science
©1925 Iowa Academy of Science, Inc.
Reilly, John F.
"Two Theorems on Annuities,"
Proceedings of the Iowa Academy of Science, 32(1), 363-365.
Available at: https://scholarworks.uni.edu/pias/vol32/iss1/71