or 
or 
ELEMENTS WHICH REDUCE VALUE 
of gradual development but of full growth and mature age, com- 
posed of numerous units, the useful life of all the units or parts of 
which is # years, it is desired to know what amount is in the 
replacement fund at any time, that fund being assumed to re- 
ceive such an increment at the end of each year that, during the 
life of each unit, this annuity, with interest, will amount to the 
original cost of this unit. 
Being composed of a large number of elements — each year 
having added new ones — the addition to it per year will be 
taken for the purpose of this illustration at one-nth of the total 
plant as it stands at the end of the nth year. 
For each dollar invested on this assumption in the first 
year, there will be $1 invested in each succeeding year, and 
for each dollar thus invested there will be # dollars of total 
investment. 
Let a represent the annual contribution to the replacement 
fund for each dollar invested. 
Assume this contribution to be available at the end of each 
year. 
Then after n years, na will be the annual contribution to the 
amortization fund for each dollar of the annual investment. 
Let m represent any number of years greater than 7. 
Let ¢ represent the interest rate expressed in hundredths, i.e., 
for 6 per cent, 2 = 0.06. 
During the first # years, after beginning the construction of 
the plant, there will be no replacements, and the replacement 
fund continues to grow. At the end of the nth year the re- 
placement requirement, assuming permanency in character and 
cost, will be $1 for each dollar of annual investment, and this 
replacement requirement will continue at this rate thereafter. 
At the end of the nth year the replacement fund will contain: 
For each dollar invested the first year: 
a(x +2)" +a(xr +4924 «+. + adollars 
HC + 2)» — 1] dollars.