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A single payment is a one-time investment of compounded at interest rate for periods that will reach a value . Or you can use the inverse, a discount factor, expressing the portion of a desired final value that must be invested to grow to in periods when invested at percent.

A uniform payment means that instead of a one-time investment, equal amounts are paid into a fund that compounds at interest rate for each of the periods. " given " gives the portion of the desired future accumulated amount that must be contributed at each time period to reach the desired goal. " given " shows the amount that must be paid to amortize an amount borrowed over the same period, with interest paid rather than earned on the outstanding balance—for example, a conventional mortgage.

The inverses of those quantities are " given " and " given ". Notice that " given " shows the value of the accumulated sum of an amount , saved in each of periods and compounding at interest rate . On the other hand, " given " is the present value of an amount that can be paid off by making payments of amount .

For example, the factor for 30 years at 5% is 15.3725, so payments of $12,000 a year would pay off a loan of 12000×15.3725 = $184,470 at that rate over that period. The factor of 66.4388 means that the same $12,000 a year contributed to, for example, a retirement savings account earning 5% would grow in 30 years to $797,265.60.

The arithmetic gradient series shows accumulations and amortizations possible by using increasing (rather than level) payments, where the amount of the payment starts at zero, but changes by dollars per year. Otherwise, this part of the Demonstration should be interpreted like the level payment accumulation and amortization factor series. Note that series of payments or contributions that start above zero and then increase at a fixed rate can be calculated as the sum of two terms—a level payment series for the initial amount and a gradient series for the increases.

Reference: T. G. Eschenbach, D. G. Newnan, and J. P. Lavelle, Engineering Economic Analysis, New York: Oxford Univ. Press, 2004.