Financial Math FM/Annuities
NOTE: This chapter assumes knowledge of capital-sigma notation for summations and some basic properties of summations and series, particular geometric series.
Geometric series 
Level payment annuities 
An annuity is a sequence of payments made at equal intervals of time. We have n periods of times . These periods could be days, months, years, fortnights, etc but they are of equal length. An annuity-immediate (also referred to an an ordinary annuity or simply an annuity) has each payment made at the end of each interval of time. That is to say, a payment of at the end of the first period, , a payment of at the end of the second period, etc.
An annuity-due has each payment made at the beginning of each interval of time.
An annuity is said to have level payments if all payments are equal. An annuity is said to have non-level payments if some payments are different from other payments. Whether an annuity has level or non-level payments is independent of whether an annuity is an annuity-due or annuity-immediate. First we'll look at the present value of an annuity-immediate with level annual payments of one using accumulation function notation.
The accumulated value of an annuity-immediate with level annual payments of one is