The rate of change in periodic cash flow in some instances is not linear.  In other words, the difference in successive cash flows is not constant.  If that difference in cash flows in successive periods is a constant percentage of the cash flow in the first of the two successive periods for all pairs of consecutive periods, the resulting series of cash flows is known as the geometric gradient series.   Thus a geometric series of cash flows is one in which the rate of increase or decrease of cash flows occurs at a faster rate than for a simple linear gradient or arithmetic series.  An example of a geometric series would be an account that is opened with some initial deposit and in which all subsequent deposits increase at a fixed percentage rate.  In problems involving geometric gradient series the constant percentage rate of change of cash flow is denoted by g; the interest rate is denoted as usual by i.

The derivation of the formula for equivalence calculation involving geometric series starts with the present worth factor.  The present worth factor is determined by summing the contributions towards the present worth made by each of the cash flows in the series.  That contribution by individual cash flows in the series is calculated using the single payment present worth  factor.  Thus we have as many contributions as there are periods.  The total present worth for all contributions is then computed using the series summing technique we have seen in previous instances.  Equations 4.21 through 4.26 detail this process.  Note the simplified expression for the present worth when the growth rate in the series, g, is equal to the interest rate, i.   In factor notation the factor for computing the present worth from a geometric gradient series is (P/A₁, g, i, N). This factor is only valid for the case of i not being equal to g.

The future worth factor for the geometric gradient series is derived from the present worth factor by multiplying it by the single payment future worth factor, (1+i)^N.   The future worth factor is shown in Equation 4.27 and, as in the case of the present worth factor, it is only valid when i is not equal to g.  For the i equal to g case, a simplified expression is used in the computation.

Unlike the other factors seen so far, no table values are available for the factors for geometric gradient series.  Computation with factors for geometric gradient series require the use of formulas.

Table 4.3 lists all the formulas described so far, for all cash flow types.  Many of the factors listed in the table are related to  each other. These relationships can be handy when not all information for directly computing a factor is available and one has to go in a roundabout fashion to compute the value of the required factor.  Some below are some of these relationships:

(F/P, i, N) = i(F/A, i, N) + 1
(P/F, i, N) = 1 - (P/A, i, N)i

(A/F, i, N) = (A/P, i, N) - i

(A/P, i, N) = i/[1 - (P/F, i, N)]

(F/G, i, N) = (P/G, i, N)(F/P, i, N)
(A/G, i, N) = (P/G, i, N)(A/P, i, N)