Engineering Economic Analysis

100 MORE INTERESTFORMULAS

Multiply Equation 4-15by (1 +i)andfactorout G, or

(1 +i)F = G[(1+i)n-l+ 2(1+i)n-2+.". +(n- 2)(1 +i)2+(n-1)(1 +i)l] (4-16)

Rewrite Equation 4-15 to' show other-terms in the series,

F=G[(1+i)n-2+". "+(n- 3)(1 +i)2+(n- 2)(1 +i)1+n- 1] (4-17)

Subtracting Equation 4-17from Equation 4-16, we obtain

F+iF - F= G[(1+it-1 + (1 +i)n-2+." .+(1 +i)2+ (1 +i)1+ 1] - nG (4-18)

In the derivation of Equation4-5, the terms inside the brackets of Equation 4-18 were shown

to equal the series compound amount factor:

(1 +")n^1

[(1 + i)n-l+ (1 +i)n-2+.".+(1+i)2+ (1+i)1+ 1]= 1.-I

Thus, Equation 4-18 becomes

.

[

(1+i)n -1

]

IF =G - nG

i

Rearranging and solving forF,wewrite

F= ~ [(1+ir - 1 - n]

(4-19)

Multiplying Equation 4-19 by the single payment present worth factor gives

P=G

[

(1 +i)n- 1 _n

] [

1

i i (1 +i)n]

=G

[

(1 +i)n- in- 1

i2(1+i)n ]

[

(1+i)n-in-1

(PjG,i,n)= i2(1+i)n ]

(4-20)

Equation 4-20 is the arithmetic gradient present worth factor. MultiplyingEquation4-19

by the sinking fund factor,we have

A= G

[

(1 +i)n- 1 _n

][

i

]

= G

[

(1 +i)n- in- 1

i i (1 +i)n- 1 i(1 +i)n- i ]

.

[

(1 +i)n- in- 1

] [

In

(AjG,I,n) = i(l +i)n -i = i - (1+i)n -1 ]

(4-21).

Equation 4-21 is the arithmetic gradient uniform series factor.