# Arithmetic Gradients

## Table of Contents

## Present Worth of an Arithmetic Gradient #

For an arithmetic gradient of costs or benefits where \(\small A\) is received in the first period, \(\small A + G\) in the second period, \(\small A + 2G\) in the third period, and so on, the present worth \(\small P\) is given by:

$$P = A \left[\frac{(1+i)^n-1}{i(1+i)^n}\right] + G \left[\frac{(1+i)^n-in-1}{i^2(1+i)^n}\right]$$

Where

- \(\small P\) is the present worth,
- \(\small A\) is the constant amount being added to in the gradient,
- \(\small G\) is the gradient or the constant increase per period,
- \(\small i\) is the interest rate, and
- \(\small n\) is the number of periods.

Note that this is just the sum of A being converted to P and G being converted to P using their respective factors.

## Uniform Series Worth of an Arithmetic Gradient #

It is also possible to convert the linearly increasing series of payments (G) to a uniform series of payments (A) using the following formula:

$$ A = G \cdot (A/G, i ,n) = G \left[\frac{1}{i} - \frac{n}{(1+i)^n - 1}\right]$$