Deriving the Ordinary Annuity Formula

When we deposit an amount, P, m times per year (at the end of each period) into an account that compounds m times per year for t years, we can find the future value of the account as follows: A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number. The sum of the first n terms of a geometric sequence can be written as: S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n-1 S n is our FV (where n is our mt),a 1 is our P,r is our Multiplying both sides of the equation by r we get: rS n = a 1 r + a 1 r 2 + a 1 r 3 + a 1 r 4 + … + a 1 r n Taking: S n – rS n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n-1 – (a 1 r + a 1 r 2 + a 1 r 3 + a 1 r 4 + … + a 1 r n ) = a 1 – a 1 r n

So: S n – rS n = a 1 – a 1 r n S n (1 – r) = a 1 (1 – r n ) Dividing both sides by (1 – r), the sum of the n terms is: Replacing S n with FV, n with mt, a 1 with P, and r with

Ordinary Annuity Formula