8- Easy Illustration of the Arithmetic Gradient.

Last Updated on September 6, 2026 by Maged kamel

Illustration of the Arithmetic Gradient.

The discussion will be about the Arithmetic gradient.

Our new subject of discussion will be the arithmetic gradient, but we will first review the relationship between P and F: P is the present value, and F is the future value. 

Review of the different types of investments.

Deposit Money to get the retirement amount at the end of your investment (P/F) relationship.

If you deposit Money in a bank at a compound interest rate i% for an amount P, you will receive a future value F for your investment.

We will find that we have different signs. P is represented by a downward arrow, or minus sign, since it is a cash-out, but when you receive an F value, it will be cash-in, and tan upward arrow draws the value The P-value in terms of F, i%, and n, and F in terms of P, i%, and n are shown in the slide.

  For the P-value, t is PP = FF*(P/F, i%, n). For the F-value, it is=F=P*(1+i)^n. P- value= F* (1+i) ^-n. The relation is based on annual compound interest i%, but if we change the annual interest to quarterly compounding. The new interest rate i2=(i1/4%) and raised to the power of 4*(number of years), or 4n. That was the relation of F with a given P, I, and n.

Deposit Money to buy a car at the end of your investment (P/F) relation.

There is another relation: for instance, if someone wishes to buy a car, they take a loan from a bank at an interest rate of i% and will make a payment of F at the end of Time n; this is from the bank’s perspective.

Cash in will be denoted by a plus sign (+), and the Money received F will be cash out, denoted by a negative sign (-).

There is another type of relationship in which someone receives Money to buy goods and repays it in installments or through a uniform series of compound-interest payments over the agreed-upon period. The P-value is cash in, with a positive sign.

The cash-out with equal amounts will be a cash-out with negative signs; its value = A. The value for A in terms of P, i, and n is A = iP (1+i)^n/((1+i)^n-1), where i is the interest rate, and n is the Time in years. The values of A are equal to those of a uniformly distributed load, with A as cash in installments in the bank account, with interest I, for some Time.

Deposit Money in a uniform series to receive insured Money at the end of an investment (A/F) relationship.

For instance, if somebody wants to receive insured Money after some Time, he will pay equal installments of value A at an interest rate of i% for n Time, starting after one year. The diagram shows this type; the A values are shown as upward cash flow, and at the end of Time, for instance, time=8 years. The F- value received at the end of year 8can be estimated based on a compound interest rate of irate of %; ;the F- value= A(F/A, i%, n), also F=A/i((1+i)^n-1).

The arithmetic gradient factors.

We will discuss a new subject: the arithmetic gradient factors.
While paying installments, these installments are not of equal value; the difference between each consecutive payment is constant. It equals G, which is the same difference between the second and third installments as well as between the third and fourth installments, as between the fourth and fifth installments.

After one year of the agreement, someone pays an installment of value A. The cash-in diagram has the shape of a trapezoid with the smaller base and the larger base.A+,4r the sketch shown where n=5 years. But if we have n years, the larger base value of the Trapezium will be A+(n-1)G. But if we have n years, the larger base value will be A + (n-1)*G.

The previous shape is a Trapezium, which can be split into two shapes.

The first shape is an equal installment diagram with a cash-in value of A for 5 years.
The second part is a triangle with a hypotenuse increasing from 0 to 4G over 5 years, as shown in the sketch, with a slope of 4G/4 = G.
If we want to estimate the future value in terms of A, i%, and n. The relation is F=(A/i)*((1+i)^n-1)). If we want to estimate A in terms of P, i, and n, the relation is A = (P*i)*(1+i)^n/((1+i)^n-1).

While for the triangular shape, the value for F is in terms of i%, n, G. F=(G/I)*(1+i)^n-1)/I)-n. P/G Find P in terms of G. The( P/G, i,n) has a relation, which is shown in the slide.

The PDF for this post can be viewed or downloaded from the attachment button.

External resource in Engineering Economy. This link illustrates different types of economies and how to make economic decisions—the Time value of Money. A good reference

The two upcoming posts are “What is Capital Recovery in the Economy?” and “A Solved Problem for EAW/ Equivalent Annual Worth.”

The third post: Arithmetic Gradient-part 2. How do you find P given I%, n, and G?