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Derive the formula for the present value of an annuity in which the periodic payment is R dollars

Derivation. The desired present value is equal to the sum of the present values of the various
payments. Using the formula for present value, P = A(1 + i)^{-1} , we see that the present value of
the payment due at the end of the first period is R(1 + i)^{-1} , the present value of the payment due
at the end of the second period is R(1 + i)^{-2},etc. The present value of the last payment, due after n
periods, is R(1 + i)^{-n }. Thus

P_{n|I} = R(1 + i)^{-1} + R(1 + i)^{-2} + .... + R(1 + i)^{-n }

or

1) P_{n|I} = R [(1 + i)^{-1} + (1 + i)^{-2} + .... + (1 + i)^{-n} ]

The terms in the right factor form a geometric progression in which (1 + i)^{-1} is both the first term
and common ratio. Using the formula for the sum of the first n terms of a geometric progression,
1) becomes

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