Engineering Economic Analysis

r

Arithmetic Gradient 99

Cash flows of this form may be resolved into two components:

A+4G

A+2GA+

r

3G 4

r

G

A+G _.-. -". 3G

A A A A A 2G

i t 1. t t t t t 0 ~ t ,

r-l-Z-3-4-S =r-l-Z-3-4-S +1-I-Z-3-4-S

p

Note that by resolving the problem in this manner, the first cash flow in the arithmetic

gradient series becomes zero. This is done so that G is the change from period to period,

and because the gradient(G)series normally is used along with a uniform series(A).We

already have an equation forP',and we need to derive an equation forP".In this way, we

will be able to write

P= P'+p"=A(P/A, i, n)+G(PfG, i, n)

Derivation of Arithmetic Gradient Factors

The arithmetic gradientis a series of increasing cash flows as follows:

(n-})G

(n-2)G

2G

o X t r

O-}-2-3~n -}-n

NOdune~O~!

The arithmetic gradientseries may be thought of as a series of individual cash flows:

(n- })G

2G

X t

0-I-Z-3-+-n-l-t +0-I-z-3-+-n-1-1 + +

Fn

(n-2)G

i

O-}-2-3-+-n-}-n +O-}-2-3-+-n-}-n

1

Fill

The value ofFfor the sum of the cash flows=FI+Fn+... +Fill+FlY,or

F=G(1 +i)n-2+2G(1+i)n-3+... +(n- 2)(G)(1+i)1+(n- l)G (4-15)